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STRUCTURAL DYNAMICS: RECENT ADVANCES Proceedings of the 6th International Conference Volume II Proceedings of the Sixth International Conference on Recent Advances in Structural Dynamics, held at the Institute of Sound and Vibration Research, University of Southampton, England, from 14th to 17th July, 1997, co-sponsored by the US Airforce European Office of Aerospace Research and Development and the Wright Laboratories, Wright Patterson Air Force Base. Edited by N,S. FERGUSON Institute of Sound and Vibration Research, University of Southampton, Southampton, UK. H.F. WOLFE Wright Laboratory, Wright Patterson Air Force Base, Ohio, USA. and C MEI Department of Aerospace Engineering, Old Dominion University, Norfolk, Virginia, USA. © The Institute of Sound and Vibration Research, University of Southampton, UK. ISBN no. 0-85432-6375 19970814 055 VV T.- , .. PREFACE The papers contained herein were presented at the Sixth International Conference on Recent Advances in Structural Dynamics held at the Institute of Sound and Vibration Research, University of Southampton, England in July 1997. The conference was organised and sponsored by the Institute of Sound and Vibration Research and co-sponsored by the Wright Laboratories, Wright Patterson Air Force Base. We wish to also thank the following for their contribution to the success of the conference: the United States Air Force European Office of Aerospace Research and Development. The conference follows equally successful conferences on the same topic held at Southampton in 1980,1984,1988,1991 and 1994. There are over one hundred papers written by authors from approximately 20 different countries, making it a truly international forum. Many authors have attended more than one conference in the series whilst others attended for the first time. It is interesting to note the change in emphasis of the topics covered. Analytical and numerical methods have featured strongly in all the conferences. This time, system identification and power flow techniques are covered by even more papers than previously. Also, there are many contributions in the field of passive and active vibration control. Papers dealing with nonlinear aspects of vibration continue to increase. These observations seem to reflect the trend in current research in structural dynamics. We therefore hope that the present series of International Conferences will play a part in disseminating knowledge in this area. We would like to thank the authors, paper reviewers and session chairmen for the part they played in making it a successful conference. My personal thanks go to the following individuals who willingly and enthusiastically contributed to the organisation of the event: Dr. H.F. Wolfe Wright Laboratories, WPAFB, USA Dr. C. Mei Old Dominion University, USA Mrs. M.Z. Strickland ISVR, University of Southampton, UK Grateful thanks are also due to many other members of ISVR who contributed to the success of the event. N.S. Ferguson Sixth International Conference on Recent Advances in Structural Dynamics Volume II Contents Page No. INVITED PAPER R.D. BLEVINS On random vibration, probability and fatigue 881 ACOUSTIC FATIGUE I 58. J. LEE and K.R. WENTZ Strain power spectra of a thermally buckled plate in random vibration 903 59. S.A. RIZZI and T.L. TURNER Enhanced capabilities of the NASA Langley thermal acoustic fatigue apparatus 919 60. I. HOLEHOUSE Sonic fatigue characteristics of high temperature materials and structures for hypersonic flight vehicle applications 935 61. M. FERMAN and H.F. WOLFE Scaling concepts in random acoustic fatigue 953 ACOUSTIC FATIGUE II 62. H.F. WOLFE and R.G. WHITE The development and evaluation of a new multimodal acoustic fatigue damage model 969 63. B. BENCHEKCHOU and R.G. WHITE Acoustic fatigue and damping technology in composite materials 981 64. D. MILLAR The behaviour of light weight honeycomb sandwich panels under acoustic loading 995 65. P.D. GREEN and A. KILLEY Time domain dynamic Finite Element modelling in acoustic fatigue design 1007 SYSTEM IDENTIFICATION II 66. U. PRELLS, A.W. LEES, M.I. FRISWELL and M.G. SMART Robust subsystem estimation using ARMA-modelling in the frequency domain 1027 67. Y.Q. NI, J.M. KO and C.W. WONG Mathematical hysteresis models and their application to nonlinear isolation systems 1043 68. M.G. SMART, M.I. FRISWELL, A.W. LEES and U. PRELLS The identification of turbogenerator foundation models from run-down data 1059 69. C. OZTURK and A. BAHADIR Shell mode noise in reciprocating refrigeration compressors 1073 70. T.H.T. CHAN, S.S. LAW and T.H. YUNG A comparative study of moving force identification 1083 71. P.A. ATKINS and J.R. WRIGHT Estimating the behaviour of a nonlinear experimental multi degree of freedom system using a force appropriation approach 1099 POWER FLOW TECHNIQUES II 72. R.S. LANGLEY, N.S. BARDELL and P.M. LOASBY The optimal design of near-periodic structures to minimise noise and vibration transmission 1113 73. J.L. HORNER Effects of geometric asymmetry on vibrational power transmission in frameworks 1129 74. M. IWANIEC and R. PANUSZKA The influence of the dissipation layer on energy flow in plate connections 1143 75. H. DU and F.F. YAP Variation analysis on coupling loss factor due to the third coupled subsystem in Statistical Energy Analysis 1151 76. S.J. WALSH and R.G. WHITE The effect of curvature upon vibrational power transmission in beams 1163 77. S. CHOI, M.P. CASTANIER and C. PIERRE A parameter-based statistical energy method for mid¬ frequency vibration transmission analysis 1179 PASSIVE AND ACTIVE CONTROL III 78. Y. LEI and L. CHEN Research on control law of active suspension of seven degree of freedom vehicle model 1195 79. M. AHMADIAN Designing heavy truck suspensions for reduced road damage 1203 80. A.M. SADRI, J.R. WRIGHT and A.S. CHERRY Active vibration control of isotropic plates using piezoelectric actuators 1217 81. S.M. KIM and M.J. BRENNAN Active control of sound transmission into a rectangular enclosure using both structural and acoustic actuators 1233 82. T.J. SUTTON, M.E. JOHNSON and S.J. ELLIOTT A distributed actuator for the active control of sound transmission through a partition 1247 83. J. RO, A. A-ALI and A. BAZ Control of sound radiation from a fluid-loaded plate using active constraining layer damping 1257 ANALYTICAL METHODS 11 84. E. MANOACH, G. DE PAZ, K. KOSTADINOV and F. MONTOYA Dynamic response of single-link flexible manipulators 1275 85. B. KANG and C.A. TAN Wave reflection and transmission in an axially strained, rotating Timoshenko shaft 1291 86. Y. YAMAN Analytical modelling of coupled vibrations of elastically supported channels 1329 87. R.S. LANGLEY The response of two-dimensional periodic structures to harmonic and impulsive point loading 1345 NONLINEAR VIBRATION III 88. H. OYANG, J.E. MOTTERSHEAD, M.P. CARTMELL and M.L FRISWELL Stick-slip motion of an elastic slider system on a vibrating disc 1359 89. R.Y.Y. LEE, Y. SHI and C. MEI A Finite Element time domain multi-mode method for large amplitude free vibration of composite plates 1375 90. P. RIBEIRO and M. PETYT Nonlinear forced vibration of beams by the hierarchical Finite Element method 1393 91. K.M. HSIAO and W.Y. LIN Geometrically nonlinear dynamic analysis of 3-D beam 1409 92. R.Y.Y. LEE, Y. SHI and C. MEI Nonlinear response of composite plates to harmonic excitation using the Finite Element time domain modal method 1423 93. C.W.S. TO and B. WANG Geometrically nonlinear response analysis of laminated composite plates and shells 1437 ANALYTICAL METHODS III 94. R.S. HWANG, C.H.J. FOX and S. McWILLIAM The free, in-plane vibration of circular rings with small thickness variations 1457 95. D.J. GORMAN Free vibration analysis of transverse-shear deformable rectangular plates resting on uniform lateral elastic edge support 1471 96. R.G. PARKER and C.D. MOTE, Jr. Wave equation eigensolutions on asymmetric domains 1485 97. A.V. PESTEREV Substructuring for symmetric systems 1501 RANDOM VIBRATION I 98. G.FUandJ. PENG Anaytical approach for elastically supported cantilever beam subjected to modulated filtered white noise 1517 99. S.D. FASSOIS and K. DENOYER Linear multi-stage synthesis of random vibration signals from partial covariance information 1533 100. CW.S.TOand Z. CHEN First passage time of multi-degrees of freedom nonlinear systems under narrow-band non-stationary random excitations 1549 101. C. FLORIS and M.C. SANDRELLI Random response of Duffing oscillator excited by quadratic polynomial of filtered Gaussian noise 1565 102. S. McWILUAM Extreme response analysis of non-linear systems to random vibration 1581 103. M. GHANBARI and J.F. DUNNE On the use of Finite Element solutions of the FPK equation for non-linear stochastic oscillator response 1597 RANDOM VIBRATION II 104. T.L. PAEZ, S. TUCKER and C. O’GORMAN Simulation of nonlinear random vibrations using artificial neural networks 1613 105. D.Z. LI and Z.C. FENG Dynamic properties of pseudoelastic shape memory alloys 1629 106. Z.W. ZHONG and C. MEI Investigation of the reduction in thermal deflection and random response of composite plates at elevated temperatures using shape memory alloys 1641 SIGNAL PROCESSING I 107. M. FELDMAN and S. BRAUN Description of non-linear conservative SDOF systems 1657 108. N.E. KING and K. WORDEN A rational polynomial technique for calculating Hilbert transforms 1669 109. D.M. LOPES, J.K. FIAMMOND and P.R. WHITE Fractional Fourier transforms and their interpretation 1685 SYSTEM IDENTIFICATION III no. J. DICKEY, G. MAIDANIK and J.M. D’ARCHANGELO Wave localization effects in dynamic systems 1701 111. P. YUAN, Z.F- WU and X.R. MA Estimated mass and stiffness matrices of shear building from modal test data 1713 112. YU. I. BOBROVNirSKn The problem of expanding the vibration field from the measurement surface to the body of an elastic structure 1719 113 M. AMABILI and A. FREGOLENT Evaluation of the equivalent gear error by vibrations of a spur gear pair 1733 ON RANDOM VIBRATION, PROBABILITY, AND FATIGUE R. D. Blevins Rohr Inc., Mail Stop 107X 850 Lagoon Drive Chula Vista, California 91910 ABSTRACT Analysis is made to determine the properties of a random process consisting of the sum of a series of sine waves with deterministic amplitudes and independent, random phase angles. The probability density of the series and its peaks are found for an arbitrary number of terms. These probability distributions are non-Gaussian. The fatigue resulting from the random vibration is found as a function of the peak-to-rms ratio. 1. INTRODUCTION Vibration spectra of aircraft components often are dominated by a relatively small number of nearly sinusoidal peaks as shown in Figure 1. The time history of this process, shown in Figure 2, is irregular but bounded. The probability density of the time history, shown in figure 3 only roughly approximates a Gaussian distribution and it does not exceed 2.5 standeird deviations. The time history of displacement or stress of these processes over a flight or a take off time can be expressed as a Fourier series of a finite number of terms over the finite sampling period T. N y = 0,nCOs(u)-ntn + <^n), 0 < in < T, Un > 0 (l) n=l Each frequency Un is a positive, non-zero integer multiple of 27r/r. The following model is used for the nature of the Fourier series: 1) the amplitudes a-n are positive and deterministic in the sense that they do not vary much from sample to sample, 2) the phases (j>n are random in the sense that they vary from sample to sample, they are equally likely to occur over the range -oo < 0n < oo. This last condition implies that the terms on the right hand side of equation (1) are statistically independent of each other. We can generate an ensemble of values of the dependent variable Y by randomly choosing M sets of N phase angles = 1,2..^), computing Y at some ffxed time from equation (1), choosing another set of phases, computing a second value of Y and so 881 on until we have a statistically significant sample of M Y's. This random phase approach, introduced by Rayleigh (1880), models a multi-frequency processes where each frequency component is independent and whose power spectral density (PSD) is known. The maximum possible (peak) value of equation (1) is the sum of the amplitude of each term (recall > 0). The mean square of the sum of independent sine waves is the sum of the mean squares of the terms. N ^peak ~ 'y ^ n=l = Na, for ai= 02 = an = CL (2a) ( 2 &) X N N Yrms = ^ f [Y anCO«(2wt„/T + <Pri)?dtn = 5 = |iVa^, for ai = 02 = an = a The peak-to-rms ratio of the sum of N mutually independent sine waves thus is, N N I rms 1/2 n=l 71=1 = (2A/')^/^, for Oi = 02 = an = o. (3a) {Sb) (4a) (46) Equation (4b) shows that the peak-to-rms ratio for an equal amplitude series increases from 2^/^ for a single term (N=l) and approaches infinity as the number of terms N approaches infinity, as shown in Figure 4. The probability of Y is zero beyond the peak value. For example, there is no chance that the sum of any four {N = 4) independent sinusoidal terms will be greater than 8^/^ = 2.828 times the overall rms value. 2. PROBABILITY DENSITY OF A SINE WAVE The probability density py (y) of the random variable Y is probability that the random variable Y has values within the small range between y and y -b dy, divided by dy. p(Y) has the units of 1/Y. Consider single a sine wave of amplitude a^, circular frequency uJri) and phcLse Y = On COs{0Jntn + 0 < < 277. (5) Y is the dependent random variable. The independent random variables are tn or <l>n- The probability density of a sine wave for equal likely phases p((l>n) = l/(27r), or equally likely times, p[tn) = 1/T, is (Bennett, 1944; Rice 1944, art. 3.10), wiy) = 77 ^(a^ - j/2) 1/2^ if <y <an\ 0 , if I 2 /I > CLn ( 6 ) 882 The probability density of the sine wave is symmetric about y = 0, i.e., pyiv) ~ PYi~y)) it is singular at y = Cn, and it falls to zero for jyl greater than an as shown in Figure 5. The characteristic function of a random variable x is the expected value of C{f) = r (7) J —CO and it is also the Fourier transform of the probability density function (Cramer, 1970, pp. 24-35; Sveshnikov, 1965; with notation of Bendat, 1958). j = is the imaginary constant. The characteristic function of the sine wave is found using equations (13) and (14) and integrating over the range 0 < X < a^. (Gradshteyn, Ryzhik, Jeffrey, 1994, article 3.753). Cn(f) = 2(7ra„)-‘ T" cos{27rfY))[l - {Y/dY = Jo(27r/a„), (8) Jo The characteristic function of a sine wave is a Bessel function of the first kind and zero order (Rice, 1944, art. 3.16). Equations (6) and (8) are starting points for determining the probability density of the Fourier series. 3. PROBABILITY DENSITY OF THE SUM OF N SINE WAVES It is possible to generate an expression for the probability density of Fourier series (equation l) with 1,2,3,to any number of terms provided the sine wave terms are mutually independent. This is done with characteristic functions. The characteristic function of the sum of N mutually independent random variables (Y = Xi -j- X 2 + -■■i- Xj\/) is the product of their characteristic functions (Weiss, 1990, p.22; Sveshnikov, pp. 124-129), c(/) = r .. r e^2-/{^.+^=+- 7 —00 7—00 N ^00 N = n / = n CM)- (9) The symbol 11 denotes product of terms. The characteristic function for the sum of N independent sine waves is found from equations (8) and (9). C{f)^ n!Li *^ 0 (27r/an), unequal an [Jo( 27 r/a)]^, ai == a2 = = a ( 10 ) The probability density of Y is the inverse Fourier transform of its characteristic function (Sveshnikov, 1968, p. 129). py{y) = r e-^^-fyc{f)df 7—00 ( 11 ) 883 By substituting equation (10) into equation (11) we obtain an integral equation for the probability density of a N-term finite Fourier series of independent sine waves (Barakat, 1974). .oo N Pviy) = 2 / cos{2Tryf) { TT Jo(27r/an) } df, ''0 n=l iV = 1,2,3... ( 12 ) If all N terms of the Fourier series have equal amplitudes a = ai — a 2 = On = a/sr, then this simplifies, py(y) = 2 r cos(2iryf)[Jo{2wfa)fdf, N = 1,2,3... (13) Jo These distributions are symmetric about y — 0 as are all zero mean, sum-of-sine-wave distributions. Figures 5 and 6 show results of numerically integrating equations (45) and (46) over interval / = 0 to / = 15a using Mathematica (Wolfram, 1995). Barakat (1974, also see Weiss, 1994, p. 25) found a Fourier series solution to equation (45). He expanded the probability density of the N term sum in a Foui'ier series over the finite interval -Ly < Y < Ly where Ly = ai + a 2 + - + The result for unequal amplitudes is , \v\<Ly. i=L n=l For equal amplitudes, ai = a 2 = a,^ = o, Ly = Na, and i=l \y\ < Na. (14) (15) Figure 6 shows that the Fourier series solution (equation 15) carried to 20 terms to be virtually identical to numerical integration of equation (13) and it compares well with the approximate solution. Note that theory requires py{\yT\ > Ly) = 0. A power series solution for equation (13) can be found with a technique used by Rice (1944, art. 16) for shot noise and by Cramer (1970) who called it an Edgeworth series. The Bessel function term in equation (13) is expressed as an exponent of a logarithm which is then expanded in a power series, [Jo(27r/a)]-^ = ex'p{N ln[Jo[2'Kaf)\), (16) = exvi-Nir'^a^f - (l/4)iV7r^o'‘/'' - (l/9)N-K^a^f + (n/192)Arx5a®/-)- = + -1 Substituting this expansion into equation (13) and rearranging gives a series of integrals, which are then solved (Gradshteyn, Ryzhik, Jeffrey, 1994, arts. 3.896, 3.952) to give a 884 power series for the probability density of the equal-amplitude N-term Fourier series sum. VY{y) = g^iFi[-2,V2,yV(2yA.,)l -( 11 r(9/2) (17) 192iV3 32iv2^ ^1/2 )^^iFi[-4,l/2,yV(2i;L.)l + -)^ \y\<Na 'PY{\y\ > = 0 and Yrms is given by equation (3b). There are two special func¬ tions in equation (17), the gamma function T and the confluent hypergeometric function iFi[n,'y,z]. These are defined by Gradshteyn, Ryzhik, and Jeffrey (1994). As N approaches infinity, the peak-to-rms (equation 4b) ratio approaches infinity, and equation (51) approaches the normal distribution, \im pY{y) = ~;^—. (18) N-*oo V^Yrms as predicted by the central limit theorem (Cramer, 1970; Lin, 1976). 4. PROBABILITY DENSITY OF PEAKS Theories for calculating the fatigue damage from a time history process generally require knowledge of the peaks and troughs in the time history. This task is made simpler if we assume that the time history is narrow band. If Y{t) is narrow band that is, that each trajectory of Y{t) which crosses zero has only a single peak before crossing the cixis again, then (1) the number of peaks equals the number of times the time history crosses the axis with positive slope, and (2) only positive peaks occur for Y{t) >0 and they are located at points of zero slope, dY{t)ldt = 0. Lin (1967, p. 304) gives expressions for the expected number of zero crossings with positive slope (peaks above the axis) per unit time for a general, not necessarily narrow band, process, ElNo+]= f ypyy(0,y)<iy (19) Jo and the probability density of the peaks for a narrow band process. = (20) In order to apply these expressions, the joint probability distribution of Y and Y must be established. The joint probability density function Pyriy^y) random variable Y and Y is the probability that Y falls in the range between y and y + dy and y falls in the range between y and y-\-dy, divided by dydy. The derivative of the sine wave Y (equation 12) with respect to time can be expressed in terms of Y, dY/dt = Y — -Gn^n sin{u)nt + 0 n) == ~ Y^, \Y\ < On. ( 21 ) 885 The joint probability density is the inverse Fourier transform of its characteristic function, / CO poo / (30) -oo J—oo The proof of equations (28), (29), and (30) can be found in Chandrasekhar (1943), Willie (1987), Weiss and Shmueli (1987), and Weiss (1994, pp. 21-26). Since the probability is symmetric about y — y = 0, Pyriv^v) ~ Pyy(~S/j“y)) only symmetric terms survive the integration. Substituting, equation (29) into equation (30) and expanding gives and integral expression for the joint probability of y and Y. Pyy(2/,y)== [ [ {JJ ^o(27ranY^/f + /|cj2)}cos(27r/iy)cos(27r/2y)d/i<i/2 (31) It is also possible to expand the joint probability of Y and Y in as double finite Fourier series. The result is: .. oo oo N I . 7 Pyriv^y) = (^)2j}cos(i7ry/Ly)cos(/:7ry/L^) aik = 1, i, /c> 0; 1/2, i — Qork = 0; 1/4, k = 0 (32) (33) The expected number of peaks per unit time and the probability distribution of the narrow band peaks is obtained by substituting this equation into equations (19) and (20) and integrating. The results are: n—1 ^0 k=0 TTUr Ly ‘ (34) y OO OO iV :/-)' +(^P)}sin(i^A/Ly) JuY J^Y where ( 1/8, i = /c = 0, _ J 1/4, i > 0, A: = 0 - 1 (l/2)[(-l)'‘- - ll/(fc 7 r)^ i = 0,fc > 0, I [(-1)* - ll/(fe7r)2, i>0.fc>0. (35) (36) If the frequencies are closely spaced so uJn^<^ and hence Ly » uLy , then one positive peak is expected once per cycle, £;[iVo+]=a;/(27r) (37) 886 and the probability density of narrow band peaks becomes, N t =0 fc =0 71=1 (38) Figure 7 shows probability density of narrow band peaks for N=2,3, and 4 equal amplitude (tti = 1) equal frequency series using equation (38). Each sum in equation (38) was carried to 40 terms. A power series solution for equation (20) can be found if all N terms in the series have equal amplitude and frequency. The result is Pa(A) = Y2 1 1 4iV 4ivy2 A^ 32Nr,i, •h... In the limit as N becomes infinite these equations become, (39) pYi-(y>y) = IttY Y Zr/i j rms*^ rms (40) Pa{A) = y2 ^ rms (41) Equation (40) is in agreement with an expression given by Crandall and Mark (1963, p. 47) and equation (41)is the Rayleigh distribution. Equations (20), (35), (38), and (41) are conservative when applied to non-narrow band processes in the sense that any troughs above the axis (points with y > 0 and dYjdt = 0 but d^Yldt^ > 0) are counted as peaks (Lin, 1967, p. 304; Powell, 1958; Broch, 1963). Equations (35), (38) and (41) can provide probability distributions for peaks of narrow band processes as a function of the number of sine waves from one to infinity and thus they model random processes with peak-to-rms ratios from 2^/^ to infinity. 5. FATIGUE UNDER RANDOM LOADING Fatigue tests are most often made with constant-amplitude sinusoidal loading. The number of cycles to failure is plotted versus the stress that produced failure and the data is often fitted with an empirical expression. MIL-HDBK-5G (1994) uses the following empirical expression to fit fatigue data, log iVy = + B 2 log{S^ - S 4 ), = 5(1 - R)^K (42) Here Nf is the cycles to failure during sinusoidal loading that has maximum stress 5 per cycle. R is the ratio of maximum to minimum stress during a cycle. R = —1 is fully 887 reversed stress cycling. Bi though B 4 are fitted parameters. With a little work, we can put this expression in the form used by Crandall and Mark (1963, p. 113). JV = cSJ*- (43) where Sd = 5(1 - - B 4 , c = 10 -®', and b = - 82 - For cycling in a time history that has non constant amplitude, Miner-Palmgren proposed that the accumulated damage is the sum of the ratios of the number of cycles at each amplitude to the allowing number of cycles to failure at that amplitude (equations 42 and 43). D = ^«(Si)/lV^(S,) (44) i where n{Si) is the number of cycle accumulated at stress amplitude Si and Nj is the number of stress cycles at this amplitude which would cause failure. Following Miles(1954) and Crandall and Mark(1963), the expected fractional damage for a random stress cycling in system with dominant cycling at frequency f in time t/, is = (45) where Pa{S) is the probability density of a stress cycle having amplitude S and Nf{S) is the number of allowable cycles to failure at this stress. Failure under random loading is expected when the expected damage is unity. Setting = 1 at time such that ftd = Nd, the inverse of the expected number of random vibration cycles to failure is This expression can be used to create a fatigue curve for random cycling given the proba¬ bility density of the random stress cycle amplitudes (p^(5)) and a fatigue curve (equation 42 with parameters Bi though B 4 and R) for sinusoidal cycling. Substituting the probability density expression for narrow band amplitude (equation 35) and for the fatigue curve(equations 42 or 43) into equation (46) and integrating, we obtain an expression for the expected number of cycles to failure as a function of the number of sine waves and their amplitudes. For N equal amplitude sine waves this is, <"->■■ - I ^1" (47) V (2 + 6)L,(l-B)®= ' + iirBi (1 - R)^Ly mBi ■ 2 ’ .1 3 1 T - irr -^ f -** ^ (1 - iV(B^(l-B)®’-B4)2, ALlil - B)2B3 ' i^7r^(£„(l - B)®° - Bif . 4L|(1 - H)2®» 888 Recall that for this case Ly = Na, the rms value is Y^ms — {l/2)Na and the peak- to-rms ratio is Peak/Yrms = y/2N (equations 2 though 4). is the generalized hypergeometric function which is a series of polynomials. It is described by Gradshteyn, Ryzhik, and Jeffrey (1994). It is also possible to establish the fatigue curve using the Rayleigh distribution (equa¬ tion 18) and the MIL-HDBK-5 fatigue curve (equation 42). The result is (2^'^r[l + 5]((1 - -BlF,[l + ^ 3 Bl 2’2’2y;2„,(i-ij)2B3« (48) Bl rr2Y,^^{i-RYB,^ _orf3 + il + 3 Bl 4 2’ 2’ 2V;2„,(1 - ii)2B3 iFi[..] is the confluent hypergeometric function which is described by Gradshteyn, Ryzhik, and Jeffrey (1994). Much of the complexity of these last two equations arises from the term B 4 which is associated with an endurance limit in the fatigue equation. That is, equation (42)predicts that sinusoidal stress cycling with stress less than 54 /( 1 —R )^=3 produces no fatigue damage. If we set B 4 = 0 to set the endurance limit to zero, then equation (48) simplifies to. This result for cycles to failure under Gaussian loading without an endurance limit is also given by Crandall and Mark (1963, p. 117). Equations (47), (48) and (49) allow us to compute the fatigue curves of a material under random loading from a fatigue curve generated under sinusoidal loading (equation 43) for narrow band random processes with any peak-to-rms ratio from 2^/^ to infinity. 6. APPLICATION Figure 8 is the MIL-HDBK-5G fatigue curve for aluminum 2024-T3 with a notch factor of Kt=4 under sinusoidal loading with various R values. The fitted curve shown in the figure, gives the following parameters for equation (42). Bl = 8.3, B 2 = -3.30, Bz = 0.66, B 4 = 8.4 889 The B 2 and B 3 are dimensionless. B 4 has the units of ksi, that is thousands of psi, and 10^^ has units of These Bi,..B 4 are substituted into equations (43), (47), (48), and (49). The fatigue curves under random loading are computed as follows, 1) the number of sine waves N is chosen and this fixes the peak-to-rms ratio from equation (3b), 2) set of values of rms stresses are chosen and for eac h the corresponding sine waves amplitudes are computed using equation (3b), a = Srmsy/VN (note that the peak stress much exceed S4=8.5 ksi), and 3)the cycles to failure are calculated from equation (47) for finite peak- to-rms ratios and equation (48) for Gaussian loading (infinite peak-to-rms). For single sine wave, the peak-to-rms ratio is 2^/^, equation 4b, and the fatigue curve interms of rms stress is adapted from the empirical data fit (equations 42, 43) by substi¬ tuting 2 ^^‘^SrTns for the stress amplitude. Nd = c(2^/25.n..(l - - B^r^ (50) where b= -B 2 and c = 10-®^ Some results are shown in Figure 9 for R=-l. 7. CONCLUSIONS Analysis has been made to determine the properties of a random process consisting of the sum of a series of sine waves with deterministic amplitudes and random phase angles. The joint probability density of the sum and its first two derivatives is determined. The probability density of the sum and narrow band peaks have been found for an arbitrary number of statistically independent sine wave terms. The fatigue cycles-to-failure resulting from these processes has been found. 1. The peak-to-rms ratio of the sum of mutually independent terms exceeds unity. If ail terms have the same peak and rms values then the peak-to-rms ratio of the series sum increases with the square root of the number of terms in the series. The probability of the series sum is zero beyond a maximum value, equal to the sum of the series amplitudes, and below the minimum value. Hence, he probability densities of the finite series, their peaks, and their envelope are non Gaussian. 3. The formulas allow the direct calculation of the probability density of the series and its peaks from its power spectra density (PSD) under the assumption that each spectral component is statistically independent. 4. The fatigue curves of a material under random loading with any peak-to-rms ratio from 2^/^ to infinity can be computed dfrectly from the fatigue curve of the material under sinusoidal loading. 890 REFERENCES Abramowitz, M. and LA. Stegun 1964 Handbook of Mathematical Functions, National Bureau of Standards, U.S. Government Printing OfRce, Washington D.C. Reprinted by Dover. Bennett, W.R., 1944 Acoustical Society of America 15, 165. Response of a Linear Rectifier to Signal and Noise. Bendat, J.S., 1958 Principles and Applications of Random Noise Theory, Wiley, N.Y. Chandrasekhar, S., 1943,Reweiys of Modem Physics, 15, 2-74. Also available in Wax, N. (ed) Selected Papers on Noise and Stochastic Processes, Dover, N.Y., 1954. Cramer, H., 1970 Random Variables and Probability Distributions, Cambridge at the Uni¬ versity Press. Crandall, S.H., and C. H. Mark 1963 Random Vibrations in Mechanical Systems, Academic Press, N.Y. Department of Defense, 1994 Metallic Materials and Elements for Aerospace Vehicle Struc¬ tures, MIL-HDBK-5G. Gradshteyn, I.S., I.M. Ryzhik, and A. Jeffrey 1994 Table of Integrals, Series, and Products 5th Ed., Academic Press, Boston. Lin, P.K., 1976 Probabilistic Theory of Structural Dynamics, Krieger, reprint of 1967 edi¬ tion with corrections. Mathematica, 1995 Ver 2.2, Wolfram Research, Champaign, Illiinois. Miles, J., 1954 Journal of Aeronautical Sciences 21, 753-762. On Structural Fatigue under Random Loading. Powell, A., 1958 Journal of the Acoustical Society of America SO No. 12, 1130-1135. On the Fatigue Failure of Structure due to Vibrations Excited by Random Pressure Fields. Rayleigh, J.W.S. 1880 Philosophical Magazine X 73-78. On the Resultant of a Large Number of Vibrations of the Same Pitch and Arbitrary Phase. Also see Theory of Sound, Vol 10, art. 42a, reprinted 1945 by Dover, N.Y.. and Scientific Papers, Dover, N.Y., 1964, Vol. I, pp. 491-496. Rice, S.O., 1944 The Bell System Technical Journal 23 282-332. Continued in 1945 24 , 46-156. Mathematical Analysis of Random Noise. Also available in Wax, N. (ed) Selected Papers on Noise and Stochastic Processes, Dover, N.Y., 1954. Shmulei, U. and G.H. Weiss 1990 Journal of the American Statistical Association 85 6-19. Probabilistic Methods in Crystal Structure Analysis. Sveshnikov, A.A, 1968 Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions Dover, N.Y., translation of 1965 edition, pp. 74, 116. 891 Tolstov, G.P., 1962 Fourier Series, Dover, N.Y., pp. 173-177. Reprint of 1962 edition. Weiss, G.H., 1994 Aspects and Applications of the Random Walk, North-Holland, Amster¬ dam. Weiss, S.H. and U. Shmulei, 1987 Physica 146A 641-649. Joint Densities for Random Walks in the Plane. Willie, L.T., 1987 Physica 141A 509-523. Joint Distribution Function for position and Rotation angle in Plane Random Walks. Wirsching, RH., T.L. Paez, and K. Ortiz 1995 Random Vibrations, Theory and Practice, Wiley-Interscience, N.Y., pp. 162-166. 892 A an Bn C{f) cih.h) E[N,] E[No^] iFi pEq f i 3 Jo k K Ly Ey m N Nf n Bviy) Py(^) pxYi^,y) S t T Y Y X aij r lij (f>n U a>n NOMENCLATURE amplitude, peak, or envelope amplitude of the nth sine wave, a^, > 0 fitted parameter in equation (42) characteristic function with parameter / joint characteristic function with parameters fi and /2 expected number of positive peaks per unit time expected number of zero crossing with positive slope per unit time confluent hypergeometric function (Gradshteyn, Ryzhik, Jeffrey, 1994, art. 9.210) generalized hypergeometric function (Gradshteyn, Ryzhik, Jeffrey, 1994, art. 9.210) parameter in Fourier transform integer index imaginary constant, Bessel function of first kind and zero order integer index complete elliptic integral of first kind, equation (33a) + ^2 + •• + sum of amplitudes ujiai + uj 2 a 2 + - • + sum of velocity amplitudes integer index number of terms in series cycles to failure integer index, n=l,2,..N cumulative probability, the integral of Py{x) from x=—co to y probability density of random parameter Y evaluated at T = a: joint probability density of X and Y evaluated at Y = y and X = x stress time, 0 < t < T length of time interval sum of N modes or terms, —Ly<Y < Ly first derivative with respect to time of Y, —Ly <Y< Ly a random variable dimensionless coefficient, equation (33) gamma function, r[(2n + l)/2] = 7r^/^2“’^(2n - 1)!! dimensionless coefficient, equation (36) Dirac delta function 3.1415926.. XiX 2 .-xi\i, product of terms phase angle of the nth sine wave, a uniformly distributed independent random variable circular frequency, a positive (non zero) real number circular frequency of the nth term, a non zero integer multiple of 27r/T 893 Figure 1 Spectrum of vibration of a component on a turbojet engine cowling. Note the finite number of distinct peaks. 894 Figure 2 Sample of the time history associated with the spectrum of Figure 1. Note the signal is bounded, irregular and quasi sinusoidal. 895 - 4.0 0 0.0 2.0 MO, OF S.D. Figure 3 Probability density of the time history of Figures 1 and 2. Note that the maximum values do not exceed plus or minus 2.5 standard deviations. Yrms * (Probability Density of Y) -3-2-10 1 2 3 Y / Yrms - Normal Distribution -Sine Wave Distribution o Equation . N=10 • Equation , N=1 Figure 5 Normal probability density (equation 18) and sine wave probability density (equa¬ tion 6) in comparison with results of numerical integration of equation (13) for N=1 and N=10. 898 Yrms* (Probability Density of Y) Yrms (Probability Density of A) -Two Sine Waves -Three Sine Waves .Four Sine Waves Figure 7 Probability density of peaks in narrow band series with equal amplitudes (ai = 02 .. = 1) and frequencies. 900 10 * in' {o' 10* 10* 10* FRTIGUE LIFE. CYCLES FIGURE 3.2.3.1.8(h). Besi-fit SI N curves for noiched. K, ~ 4.0 of 2024-T3 aluminum alloy sheet, longitudinal direction. Figure 8 Fatigue curves for notched 2024-T3 aluminum alloy with Kt=4. MI1-HDBK-5G (1994, p. 3-115) 902 ACOUSTIC FATIGUE I Strain Power Spectra of a Thermally Buckled Plate in Random Vibration Jon Lee and Ken R. Wentz Wright Laboratory (FIB) Wright-Patterson AFB, OH 45433, USA Abstract Several years ago, Ng and Wentz reported strain power spectra measured at the mid-point of a buckled aluminum plate which is randomly excited by an electrodynamic shaker attached to the clamped-plate boundary fixture. We attempt to explain the peculiar features in strain power spectra by generating the corresponding power spectra by the numerical simulation of a single-mode equation of motion. This is possible because the essential dynamics takes place in the frequency range just around and below the primary resonance frequency. 1. Introduction For high performance military aircraft and future high-speed civil transport planes, certain structural skin components are subjected to very large acoustic loads in an elevated thermal environment [1]. This is because high-speed flights call for a very powerful propulsion system and thereby engendering acoustic loads in the anticipated range of 135-175 dB. More importantly, because of the aerodynamic heating in hypersonic flights and the modern trend in integrating propulsion sub-systems into the overall vehicular configuration, some structural components must operate at high temperatures reaching up to 1300°F. Hence, the dual effect of thermal and acoustic loading has given rise to the so-called thermal-acoustic structural fatigue [2,3]. Generally, raising the plate temperature uniformly but with an immovable edge boundary constraint would result in thermal buckling, just as one observes flexural buckling as the inplane stress along plate edges is increased beyond a certain critical value. This equivalence has been recognized [4,5] and exploited in previous analytical and experimental investigations of the thermal- acoustic structural fatigue [6,7,8]. An experimental facility for thermal- acoustic fatigue, termed the Thermal Acoustic Fatigue Apparatus, was constructed at the NASA Langley Research Center in the late 80’s. Under the acoustic loading of 140-160 dB, Ng and Clevenson [9] obtained some strain measurements of root-mean-square value and power spectral density (PSD) on an aluminum plate heated up to 250^. Later, Ng and Wentz [10] have repeated the heated Aluminum plate experiment but by randomly exciting the clamped-plate boundary fixture by a shaker, and thereby recovering similar strain measurements. It should be noted that Ng and his colleagues [7,9,10] were the first to achieve sufficient plate heating to induce thermal buckling and thus observe the erratic snap-through under the acoustic or shaker excitations. Here, by erratic 903 we mean that a snap-through from one static buckled position to another takes place in an unpredictable fashion. We reserve the adjective chaotic for a snap- through occurring under the deterministic single-frequency forcing [11,12]. It has already been observed that certain of the buckled plate experiment can be explained, at least qualitatively, by a single-mode model of plate equations. This is also validated by a theoretical analysis. Indeed, we showed that a single¬ mode Fokker-Planck formulation can predict the high-temperature moment behavior and displacement and strain histograms of thermally buckled plates, metallic and composite [13,14]. In retrospect, a single-mode model has proven more useful than originally intended. That is, the single-mode Fokker-Planck formulation of an isotropic plate lends itself to predicting certain statistics of composite plates which are simulated by multimode equations or tested experimentally by multimode excitations. For a refined and more quantitative comparison, one must inject more realism into dynamical models by including the multimode interactions. However, before giving up the single-mode plate equation, there is an important problem that this simple model is well suited for investigation. That is, prediction of the strain PSD measurement by Ng and Wentz [10]. As we shall see in Sec. 4, the strain PSD of a thermally buckled plate exhibits a strong spectral energy transfer toward zero frequency, and thereby saturating frequency range well below the primary resonance frequency. This downward spectral energy transfer can be modeled quite adequately by the single-mode plate equation without necessitating multimode interactions. 2. Equation of motion for the aluminum plate experiment By the Galerkin procedure, the von Karman-Chu-Herrmann type of large- deflection plate equations give rise to infinitely coupled modal equations [15]. However, much has been learned from a prototype single-mode equation for displacement^ [13,14]. q + Pq + k„{l-s)q + aq^ = g„ + g{t), (1) where the overhead dot denotes d/dt and the viscous damping coefficient is P = 2^^ with damping ratio ^. For the clamped plate, we have ;i„=f(r‘'+2rV3 + l), s=rji+(1 -M) (1+(r^+ ir^) /6], a = ^{(7^+r'^+2^i) + |(i-/i^)[T(r^+r'^) +^(.r+r''T^ + (r+47'‘)‘^ + (47+ 7"‘r^]}, &= (r‘'+2r^/3 +i)Sjj6. Note that the expressions for s and g„ are specific to the typical temperature 904 variation and gradient profiles assumed in Ref. [15]. Here, 7 = b/a is the aspect ratio of plate sides a and b, and fi is Poisson's ratio. The uniform plate temperature is measured in units of the critical buckling temperature. The maximum temperature variation on the mid-plate plane is denoted by and TJ5g is the maximum magnitude of temperature gradient across the plate thickness, where 5^ and 5^ are scale factors. Hence, 0 signifies no temperature variation over the mid-plate plane, and 0 zero temperature gradient across the plate thickness. Finally, g{t) denotes the external forcing. The parameter s represents thermal expansion due to both the uniform plate temperature rise above room temperature and temperature variation over the mid-plate uniform temperature. The combined stiffness k^(l - s)q consists of the structural stiffness k^q and thermal stiffness -sk^q , which cancel each other due to the sign difference. It is positive for 5 <1, then Eq. (1) has the form of Duffing oscillator with a cubic term multiplied by a , which represents geometric nonlinearity of membrane stretching. For s >1 Eq. (1) reduces to the so-called buckled-beam equation of Holmes [11] with a negative combined stiffness. In contrast, denotes thermal moment induced by a temperature gradient across the plate thickness; hence, it appears in the right-hand side of Eq. (1) as an additional forcing. The interplay of the terms involving 5 , a, and g^ can best be illustrated by the potential energy [15] U{q) = -goq + k^(X-s)(fn -H a . (2) Fig. 1 shows that V{c^ is symmetric when g„ = 0. For s<l it has a single well which splits into a double well as s exceeds unity. Note that the distance between the twin wells increases as for large s (Fig. 1(b)). This interpretation is valid approximately for go>^- That is, a positive g^ lowers the positive side potential (^> 0 ) and raises the negative side potential {q<0), and thereby rendering the potential energy asymmetric. U(q) ^(^1) Fig. 1 Potential energy, (a) s<V, (b) ^ >1, w = ^k^(s - l)/a , d = - l))V4a. (- ^.= 0 ; --- 5 „> 0 ) 905 It must be pointed out that Eq. (1) is dimensionless and involves explicitly only 7 and /i. For the aluminum plate experiment [10], 7=10 in./8 in. and so that k^=Q3.9l and a=85.33. If we further assume 5^- 0 for simplicity, the thermal parameter reduces to s=T^. Previously, Eq. (1) was used for the investigation of stationary Fokker-Planck distribution which involves only the ratio p!F, where F is the constant power input [13,14]. Hence, nondimensionalization has indeed spared us from specifying in detail other plate parameters. Things are however different in numerical simulation because we must know the characteristic scales to correctly interpret time- dependent solutions. By retracing the derivation, we find that the dimensionless quantities in Eq. (1) are (Eq. (IV. 1) in Ref. [1]) q-qlh, t-t/t*, g = glg*, (3) where the overhead bar denotes the physical quantity. Here, the plate thickness h, t*={b/Kf.y[ph/D, and g^=p(h/t*)^ are the characteristic length, time, and force, respectively (p = mass density, D=Eh^f\2{\~p}), £= Young’s modulus of elasticity), as listed in Table L We now rewrite Eq. (1) with the numerical coefficients (Table I). _ q + 0.0978? + 23.91(1 -s)q + 85.33?^ = (4) where g{t) has the unit of psi. Table I. Parameter values for the aluminum plate experiment 7, « 10in./8in., VoX 0.01, 23.910, 85.332 5.. 5, p o h, f*, g'^- 0.05 in., 3.305 lO'^sec.^, 5.806 1 O'’psi^ (+) p = 0.098 Ib/in^ and E =1.03 10’ psi. 3. Monte-Carlo simulation Because of 5 =0, Eq. (4) has the standard form of Duffing (s<l) and Holmes (s>l) oscillators. In stead of a single frequency for forcing g{t) [11, 12], in Monte-Carlo simulation all forcing frequencies are introduced up to a preassigned maximum so that forcing represents a plausible physical realization. Of course, particular interest here is a constant PSD. We shall begin with generation of a time-series for random processes with such a PSD. 3.1 Random forcing time-series We adopt here the procedure for generating a time-series of Shinozuka and Jan [16], which has been used for a oscillator study [17] and extensively for structural simulation applications by Vaicaitis [18,19]. Since it relies heavily on the discrete fast Fourier transforms, such as FFTCF and FFTCB subroutines of the IMSL library, it is more expedient to describe the procedure operationally rather than by presenting somewhat terse formulas. Let us 906 introduce Nj: frequency coordinates which are equally spaced in the band width A/=/^ 3 x/A^^. Now, the task is to generate a time-series of total time T that can resolve up to . Assume T is also divided into time coordinates with the equal time interval At=T/Nj, From the time-frequency relation r=l/A/, we find ^At. If we choose N Nf = ( 5 ) is the Nyquist frequency, consistent with our original definition of the upper frequency limit of resolution. A random time-series with a constant PSD can be generated in the following roundabout way. We begin by assuming that we already have a forcing power spectrum ^g{f) of constant magnitude over [0,/njax]- Such a PSD may be represented by a complex array A„= VC exp(~27rz0„) (n = 1, Nf), where takes a random value distributed uniformly in [0, 1]. Clearly the magnitudes of are C, hence We then enlarge the complex array A„ by padding with zeros for n =Ny+l, and Fourier transform it to obtain a complex array B^{n= N^). The random time-series for is now given by the real part of = Real part of (n = 1, N^) (6) As it turns out, when g „ is padded with zeros for the imaginary components and Fourier transformed, we recover the original array A„ (w=l, Nf) with Since the spectrum area is nothing but total forcing power <g^> (say, in psi^), we can relate C with the variance <g^> of pressure fluctuations, which is often expressed by the sound pressure level (SPL) in dB, according to SPL=10 log<g2 >/p2, where p=2,9 10*^ psi. Hence, c = -. (7) /max Here, Eqs. (6) and (7) defined heuristically are meant to explain the corresponding formulas (2) and (12) in Ref [18]. For the numerical simulation we first note that the resonance frequency of Eq. (1) is /^=-y^/27r« 0.778 for s ~ 0. This gives the dimensional resonance frequency fjt*~235.5 Hz which is somewhat larger than the experimental 217.7 Hz (Fig. 3(a)). As shown in Table II, we assign (~9/r) because the electrodynamic shaker used in the experiment [10] has the upper frequency limit 2000 Hz. Table II. Dimensionless parameter values for the numerical simulation at, N, _ 7, 8192, 4096 At, T 0.071. 585 907 3.2 Displacement power spectrum Under a random time integration of Eq. (4) yields a time-series for q^. We first comment on the time integration. Although there are special solvers [17,20] proposed for stochastic ordinary differential equations (ODEs), we shall use here the Adams-Bashforth-Moulton scheme of Shampine and Gordon [21], which has been implemented in DEABM subroutine of the SLATEK library. Although DEABM has been developed for nonstochastic ODEs, its use for the present stochastic problem may be justified in part by that one recovers linearized frequency response functions by the numerical simulation (Sec. 3.4). Obviously, this does not say anything about the strongly nonlinear problem in hand, and it should be addressed as a separate issue. In any event, DEABM requires the absolute and relative error tolerances, both of which are set at no larger than 10“^ under the single-precision algorithm for time integration. Note that actual integration time steps are chosen by the subroutine itself, commensurate with the error tolerances requested. Recall that is updated at every time interval Ar, and we linearly interpolate the forcing value within A?. We begin time integration of Eq. (4) from the initial configuration at the bottom of the single-well potential, ^(0)=p(0)=0, for 5' <1 and the positive side double-well potential, q{0)=^kj,s~\)/a andp(0)=0, for j >1. And we continue the integration up to T. By Fourier transforming time-series q^, we obtain displacement power spectrum 0^(/). This process of integrating and transforming is repeated over three contiguous time ranges of 7, and the successive PSDs are compared for stationarity. Since it is roughly stationary after three repetitions, we report here only the PSD of the third repetition. From the stationary input-output relation [22] where is the magnitude of system frequency response function, we write = (S) Since O (/)=C, the and would have a similar functional s dependence upon /, Hence, we call them both the displacement PSD. 3.3 Strain power spectrum Although displacement is the direct output of numerical simulation, one measures strain rather than the displacement in plate experiment. At the present level of plate equation formulation, the strain e is given by the quadratic relation e = + C^q + C2(f‘ ■> (9) where C,- are given at the middle {x/a = y/b =1/2) of a clamped plate as follows (Appendix D of Ref [13]) 908 c = (/+2yV3+l)?;g, 3(1+At)(y"+1) {l-li}- q = 8 y^ 3 ’ r 32 fy^ 5n (l-/iyV4) ^ 9 [2 16 2(y+y-‘f (y+4y-‘)^ (4y + y‘fJ For we have C^=0, Q =4.17, and Q =2.77 (Table I). Hence, Eq. (9) engenders only the linear and quadratic transformations, but no translation. In any event, translation has no effect on the spectral energy contents. By Fourier transforming time-series (n=l, A^^), we obtain strain power spectrum . Although the forcing PSD is not constant, one computes the forcing spectrum ratio as in Eq. (8) and call it the magnitude square of strain frequency response function for the lack of a better terminology. 3.4 The linear oscillators For the pre-buckled (5 <1) linear oscillator (a= 0) we rewrite Eq. (4) in standard form _ q + + 0)1(1 -s)q = (10) where col=k^, and obtain I H^(f)^ = [(0)1(1 -s)- + (An^co^ffT'- (11) As shown in Fig. 2(a), the numerical simulation of Eq. (10) recovers as given by Eq. (11) over the entire frequency range. Although the simulation of Fig. 2(a) was carried out with SPL=130 dB, it does not depend on SPL since Eq. (10) is linear. Physically speaking, Eq. (10) oscillates in a single-well potential (Fig. 1(a)). Since the potential energy has two wells (Fig. 1(b)) for s >1, we linearize Eq . (1) arou nd the positive side potential well by the transformation q=q'+^k^{s-l)la . Hence, the corresponding linear oscillator is (a) (b) Fig. 2 Linear frequency response functions, (a) Displacement; (b) Strain (j = 0; -Numerical simulation; • Eq. (U)) 909 q' + 2^0)^ q’ + Icolis -V)q' = (12) s In parallel to Eq. (11), the frequency response function of a post-buckled plate I = [(^-colis - 1) - 4;rV^)2 + . (13) The resonance frequency f=co^^2{s-l)/27J: of a post-buckled (s >1) plate should be compared with f=co^^2{\~s)/2n of the pre-buckled (j <1) plate. Now, for the linear oscillators we see that is also given by Eq. (11) and (13) for ^ <1 and >1, respectively (Fig. 2(b)). This is because the spectral energy distribution is not at ail affected by a linear transformation. 4. Displacement and strain power spectra As we shall see in Sec 4.1, the experimental strain PSD exhibits downward spectral energy transfer toward zero frequency, so that there is a considerable spectral energy buildup below the resonance frequence as SPL is raised. Moreover, it also involves an upward spectral energy transfer which then contributes to both the increased resonance frequency and broadened resonance frequency peak. Since spectral energy transfers take place around and below the primary resonance frequency, it is possible to depict the downward and upward spectral energy transfers by the numerical simulation of Eq. (4) without necessitating multimode interactions. We shall first discuss the characteristic features of experimental strain PSDs. 4.1 Experimental strain PSD Of the spectra reported in Ref. [10], we consider the following two sets. One is the nonthermal set (^=0) consisting of two PSDs of small and large SPLs. The other is the post-buckled set (5=1.7) of four PSDs. For the convenience of readers, we have reproduced in Figs. 3 and 4 the selected PSDs from Ref. [10] by limiting the upper frequency to 600 Hz, and the pertinent data are summarized in Table m. Table in. Strain power spectra of experiment and numerical simulation Fig. 4(a) Fig. 6a Fig. 4(b) Fig. 6b Fig. 4(c) Fig. 6c Fig. 4(d) Fig. 6d (*) Fig. 7 Fig. 8 Fig. 9 Fig. 10 (*) Computed from the acceleration a measured in units of g. 910 Fig. 4 Experimental strain PSD =1.7). (a) 130.1dB; (b) 142dB; (c) 151.5dB; (d) 154.6dB The following observations are drawn from the experimental PSDs. First, for the nonthermal plate Figure 3fa) : Compare the measured strain fr-2\l Hz with the theoretical displacement235 Hz of Eq. (4). Note that a small spectral energy peak is found at 467 Hz which is about twice (-2.15) the strain value. 911 Figure Sfb) : With SPL~150 dB the strain increases to 240 Hz and the spectral width at the half resonance peak has nearly doubled. The spectral energy buildups at zero and 515 Hz are more noticeable than in Fig. 3(a). Again, 515 Hz is about twice (-2.15) the primary strain f,. (a) (b) f f Fig. 5 Numerical simulation results under .y=0 and SPL=130 dB. (a) Displacement (-simulation,* Eq. (11)); (b) Strain (-simulation, • Eq. (11)); (c) PSD averaged over 12 frequency intervals (-displacement, —• — strain); (d) Strain PSD. Next, for the thermally buckled plate Figure 4(a) : The primary strain fr=227 Hz should be compared with the theoretical displacement /^=279 Hz of Eq. (13). A second spectral energy peak is found at 537 Hz, much larger than twice (-2.37) the primary strain /^. Figure 4rb) : Here, the spectral energy buildup is most significant at zero frequency. Besides, there appear two spectral energy humps at 100 and 183 Hz, below the primary strain = 227 Hz of Fig. 4(a). Discounting the zero- frequency spectral peak, PSD may be approximated by a straight line in the semi-log plot, hence it is of an exponential form up to 400 Hz. Figure 4rc) : The zero-frequency peak is followed by a single spectral energy hump at 115 Hz. Again, PSD can be approximated by a straight line and its slope is roughly the same as in Fig. 4(b). Figure 4rd) : A major spectral energy peak emerges at 130 Hz, followed by a minor one at 350 Hz. Theoverall spectral energy level is raised so that the magnitude of PSD ranges over only two decades in the figure. In Figs. 4(b)-(d) we have ignored the spectral energy peaks at around 500 Hz, for they are not related to the first plate mode under consideration. This is further supported by the simulation evidence to be discussed presently. 4.2 Numerical simulation results After choosing .y = 0 or 1.7, we are left with SPL yet to be specified. o Ideally, one would like to carry out the numerical simulation of Eq. (4) by using SPL of the plate experiment 'm-z (Table III) and thus generate strain ^ PSDs which are in agreement with Figs. 3 and 4. Not surprisingly, the _4 reality is less than ideal. An obvious reason that this cannot be done is that the forcing energy input is fed into all f plate modes being excited in Hg. 6 PSD averaged over 12 frequency experiment, whereas the forcing (j=0, SPL=138dB) energy excites only one mode in the - displacement; -•-strain numerical simulation. Consequently, SPL for the numerical simulation should be less than the experimental SPL, but we do not know a priori how much less. We therefore choose a SPL to bring about qualitative agreements between the single-mode simulation and multimode experiment. As anticipated, the simulation SPLs (Table HI) are consistently smaller than the experimental values. The numerical simulation results are shown in Figs. 5-6 for 5 = 0 and Figs. 7-10 for s =1.7. Actually each figure has four frames, denoted by (a)-(d). First, frames (a) and (b) depict and Since they are very jagged at large SPLs, we average the spectral energy over 12 frequency intervals and present both of the smoothed-out frequency response functions in the same frame (c). Lastly, frame (d) shows Og(/) itself Since there is no qualitative difference between <E>g(/) and we shall call them both the strain PSD. We present all four frames (a)—(d) of Figs. 5 and 7, but only the frame (c) of Figs. 6, 8, 9 and 10 here for the lack of space. First, for the nonthermal plate Figure 5 : The simulated is closely approximated by Eq. (11) with f = 236 Hz. Note that is also approximated by Eq. (11) for all frequencies 913 Fig. 7 Numerical simulation results under j=1.7 and SPL=129 dB. (a) Displacement (-simulation,* Eq. (13)); (b) Strain (-simulation, • Eq. (13)); (c) PSD averaged over 12 frequency intervals (-displacement, —• strain); (d) Strain PSD. but zero and 476 Hz, where the strain spectral energy piles up due to the quadratic transformation (9). Since 476 Hz is nearly twice (-2.02) the primary /^, strain spectral energy buildups are due to the sum and difference of two nearly equal frequencies, ± / 2 , where/i==/ 2 ^/^. Figure 6 : The primary strain is shifted slightly upward to 253 Hz and the spectral width at half resonance peak is 50% wider than that of Fig. 5(c). The spectral energy builds up at 525 Hz which is roughly twice (-2.08) the /^. At SPL=138 dB we find that the strain spectral energy hump at 525 Hz is about 2 decades below the resonance frequency peak, as was in Fig. 3(b). Now, for the thermally buckled plate Figure 7 : The simulated and are weU approximated by Eq. (13) around /^=270 Hz which is a litde below the linearized /^=279 Hz. Unlike in Fig. 5 for 5=0, both and l/7^(/)F show spectral energy building up significantly near zero and 543 Hz which is twice (-2.01) the /^. 914 2p —!- 1 -!- 1 - 1 -T—n- 1 -r ^_1-1-1-<-1->-1-^-1 0 300 600 f Fig. 8 PSD averaged over 12 frequency intervals (j =1.7, SPL=138 dB) -displacement; —• — strain Note that in Fig. 7(a) the spectral energy hump at 543 Hz is about 3 decades below the primary frequency peak, as was in Fig. 4(a). Figure 8 : After a large zero-frequency peak, two spectral energy humps appear at 131 Hz and 236 Hz. Note that the ratios of these frequencies to the /, (131/279 -0.47 and 236/279 - 0.85) are comparable with the same ratios (100/227 -0.44 and 183/227 - 0.81) found in Fig. 4(b). Excluding the zero-frequency peak, the overall strain PSD is a straight line, hence of an exponential form, as in Fig. 4(b), Figure 9 : The zero-frequency spectral peak is followed by a single major energy hump at 154 Hz. The ratio of this to the (154/279 -0.56) is somewhat larger than the ratio (115/227 -0.51) in Fig. 4(c). The strain PSD can also be approximated by a straight line over the entire frequency range, and Figs. 8 and 9 seem to have the same slope when fitted by straight lines. Figure 10 : The spectral magnitude of is larger than that of in the frequency range above 300 Hz. The choice of SPL=146 dB was based on that the PSD magnitude around 300 Hz is about 2 decades below the main spectral peak magnitude at 180 Hz, thus emulating Fig. 4(d). All in all, by numerical simulations we have successfully reproduced the peculiar features in the two sets of strain PSDs observed experimentally under 5 = 0 and 1.7. Fig. 9 PSD averaged over 12 frequency intervals (s =1.7, SPL=143 dB) -displacement; —•—strain Fig. 10 PSD averaged over 12 frequency intervals (j =1.7, SPL=146 dB) -displacement; —•—strain 915 5. Concluding remarks At low SPL the nonthermal {s= 0) and post-buckled (^=1.7) plates appear to have a similar PSD. However, this appearance is quite deceptive in that the nonthermal plate motion is in a single-well potential, so that PSD does not change qualitatively as SPL is raised. On the other hand, the trajectory of a post-buckled plate is in one of the two potential energy wells when SPL is very small. However, as we raise SPL such a plate motion can no longer be contained in a potential well, and hence it encircles either one or both of the potential wells in an erratic manner. This is why the experimentally observed and numerically simulated strain PSDs of a post-buckled plate exhibit qualitative changes with the increasing SPL, and thereby reflect the erratic snap-through plate motion. A quantitative analysis of snap-through dynamics will be presented elsewhere. Lastly, we wish to point out that a PSD of straigh-line form in the semi-log plot was observed in a Holmes oscillator when trajectories are superposed randomly near the figure-eight separatrix [23]. Acknowledgments Correspondence and conversations with Chung Fi Ng, Chuh Mei, Rimas Vaicaitis, and Jay Robinson are sincerely appreciated. We also wish to thank the referees for their helpful suggestions to improve the readability of this paper. References 1. Lee, J., Large-Amplitude Plate Vibration in an Elevated Thermal Environment, WL-TR-92-3049, Wright Lab., Wright-Patterson AFB, OH, June, 1992. 2. Jacobson, M.J. and Maurer, O.F., Oil canning of metallic panels in thermal-acoustic environment, AIAA Paper 74-982, Aug., 1974. 3. Jacobson, M.J., Sonic fatigue of advanced composite panels in thermal environments, J. Aircraft, 1983, 20, 282-288. 4. Bisplinghoff, R.L. and Pian, T.H.H., On the vibrations of thermally buckled bars and plates, in Proc. 9th Inter. Congr. of Appl. Mech., Brussels, 1957, 7, 307-318. 5. Tseng, W.-Y., Nonlinear vibration of straight and buckled beams under harmonic excitation, AFOSR 69-2157TR, Air Force Office of Scientific Research, Arlington, VA, Nov., 1969. 6. Seide, P. and Adami, C., Dynamic stability of beams in a combined thermal-acoustic environment, AFWAL-TR-83-3072, Flight Dynamics Lab., Wright-Patterson AFB, OH, Oct., 1983. 7. Ng, C.F., Nonlinear and snap-through responses of curved panels to intense acoustic excitation, /. Aircraft, 1989, 26, 281-288. 8. Robinson, J.H. and Brown, S.A., Chaotic structural acoustic response of a milled aluminum panel, 36th Structures, Structural Dynamics, and 916 Material Conference, AIAA-95-1301-CP, New Orleans, LA, 1240-1250, Apr. 10-13, 1995. 9. Ng, C.F. and Clevenson, S. A., High-intensity acoustic tests of a thermally stressed plate, J, Aircraft, 1991, 28, 275-281.. 10. Ng, C.F. and Wentz, K.R., The prediction and measurement of thermo¬ acoustic response of plate structures, 31st Structures, Structural Dynamics, and Material Conference, AIAA-90-0988-CP, Long Beach, CA, 1832-1838, Apr. 2-4,1990. 11. Holmes, P., A nonlinear oscillator with a strange attractor, Phil. Trans., Roy. Soc. of London, 1979, 292A, 419-448. 12. Dowell, E.H. and Pezeski, C., On the understanding of chaos in Duffings equation including a comparison with experiment, J. Appl. Mech., 1986, 53, 5-9. 13. Lee, J., Random vibration of thermally buckled plates: I Zero temperature gradient across the plate thickness, in Progress in Aeronautics and Astronautics, 168, Aerospace Thermal Structures and Materials for a New Era, Ed. E.A. Thornton, AIAA, Washington, DC, 1995. 41-67. 14. Lee, J., Random vibration of thermally buckled plates: n Nonzero temperature gradient across the plate thickness, to appear in J. Vib. and Control, 1997. 15. Lee, J., Large-amplitude plate vibration in an elevated thermal environment, Mech. Rev., 1993, 46, S242-254. 16. Shinozuka, M. and Jan, C.-M., Digital simulation of random processes and its applications, J. Sound and Vib.1912, 25, 111-128. 17. Chiu, H.M. and Hsu, C.S., A cell mapping method for nonlinear deterministic and stochastic systems - Part II: Examples of application, J. Appl. Mech., 1986, 53, 702-710. 18. Vaicaitis, R., Nonlinear response and sonic fatigue of national aerospace space plane surface panels, J. Aircraft, 1994, 31, 10-18. 19. Vaicaitis, R., Response of Composite Panels Under Severe Thermo- Acoustic Loads, Report TR-94-05, Aerospace Structures Information and Analysis Center, Wright-Patterson AFB, OH, Feb., 1994. 20. Kasdin, N.J., Runge-Kutta algorithms for the numerical integration of stochastic differential equations, J. Guidance, Control, and Dynamics, 1995,18, 114-120. 21. Shampine, L.F. and Gordon, M.K., Computer solution of ordinary differential equations, 1975, Freeman, San Francisco. 22. Lin, Y.K., Probabilistic theory of structural dynamics, Robert E. Krieger Publishing, 1976, Huntington, NY. 23. Brunsden, V., Cortell, J. and Holmes, P.J., Power spectra of chaotic vibrations of a buckled beam, J. Sound and Vib., 1989,130, 1-25. 917 918 ENHANCED CAPABELITIES OF THE NASA LANGLEY THERMAL ACOUSTIC FATIGUE APPARATUS Stephen A. Rizzi and Travis L. Turner Structural Acoustics Branch NASA Langley Research Center Hampton, VA 23681-0001 ABSTRACT This paper presents newly enhanced acoustic capabilities of the Thermal Acoustic Fatigue Apparatus at the NASA Langley Research Center. The facility is a progressive wave tube used for sonic fatigue testing of aerospace structures. Acoustic measurements for each of the six facility configurations are shown and comparisons with projected performance are made. INTRODUCTION The design of supersonic and hypersonic vehicle stmctures presents a significant challenge to the airframe analyst because of the wide variety and severity of environmental conditions. One of the more demanding of these is the high intensity noise produced by the propulsion system and turbulent boundary layer [1]. Complicating effects include aero-thermal loads due to boundary layer and local shock interactions, static mechanical preloads, and panel flutter. Because of the difficulty in accurately predicting the dynamic response and fatigue of structures subject to these conditions, experimental testing is often the only means of design validation. One of the more common means of simulating the thermal-vibro-acoustic environment is through the use of a progressive wave tube. The progressive wave tube facility at NASA Langley Research Center, known as the Thermal Acoustic Fatigue Apparatus (TAFA), has been used in the past to support development of the thermal protection system for the Space Shuttle and National Aerospace Plane [2]. It is presently being used for sonic fatigue studies of the wing strake subcomponents on the High Speed Civil Transport [3]. The capabilities of the TAFA were previously documented by Clevenson and Daniels [4]. The system was driven by two Wyle WAS 3000 airstream modulators which provided an overall sound pressure level range of between 125 and 165 dB and a useful frequency range of 50-200 Hz. A 360 kW quartz lamp bank provided radiant heat with a peak heat flux of 54 W/cm^. A schematic of the facility is shown in Figure 1. Representative spectra and coherence plots are shown in Figures 2 and 3. Since that time, the facility has undergone significant enhancements designed to improve its acoustic capabilities; the heating capabilities were not changed. The objectives of the enhancements were to increase the maximum overall sound pressure level (OASPL) to 178 dB, increase the frequency bandwidth to 500 Hz and improve the uniformity of the sound pressure field in the test section. This paper 919 documents the new capabilities of the TAFA and makes comparisons with the projected performance. Figure 1: Schematic of the old TAFA facility. Figure 2: Test section spectra of the Figure 3: Test section coherence of the old TAFA facility. old TAFA facility. FACILITY DESCRIPTION In order to meet the design objectives, extensive modifications were made to the sound generation system and to the wave tube itself. A theoretical increase of 6 dB OASPL was projected by designing the system to utilize eight WAS 3000 air modulators compared to the two used in the previous system. A further increase of nearly 5 dB was expected by designing the test section to accommodate removable water-cooled insert channels which reduced its cross- sectional area from 1.9m x 0.33m to 0.66m x 0.33m. The frequency range was increased through the use of a longer horn design with a lower (15 Hz vs. 27 Hz in the old facility) cut-off frequency, use of insert channels in the test section to shift the frequency of significant standing waves above 500 Hz, and design of facility sidewall stmctures with resonances above 1000 Hz. The uniformity of the sound pressure field in the test section was improved through several means. A new, smooth exponential horn was designed to avoid the impedance mismatches of the old design. To minimize the effect of uncorrelated, broadband noise (which develops as a byproduct of the sound 920 generation system), a unique design was adopted which allows for the use of either two-, four-, or eight-modulators. When testing at the lower excitation levels for example, a two-modulator configuration might be used to achieve a lower background level over that of the four- or eight-modulator configurations. In doing so, the dynamic range is extended. Lastly, a catenoidal design for the termination section was used to smoothly expand from the test section. Schematics of the facility in the three full test section configurations are shown in Figures 4-6. In the two-modulator configuration, the 2 x 4 transition cart acts to block all but two of the eight modulators. The facility is converted from the two- to four-modulator configuration by the removal of the 2 x 4 transition cart and connection of two additional modulators. In doing so, the modulator transition cart slides forward and thereby maintains the continuous exponential expansion of the duct. In the four-modulator configuration, the 4 X 8 transition cart acts to block the two upper and two lower modulators. Removal of this component and connection of the four additional modulators converts the facility to the eight-modulator configuration. Again, the continuous exponential expansion is maintained as the modulator transition cart slides forward. Figure 4: Two-modulator full test section configuration. 921 Figure 5: Four-modulator full test section configuration. Figure 6: Eight-modulator full test section configuration. Schematics of the three reduced test section configurations are shown in Figures 7-9. In these configurations, the horn cart is discarded and the horn transition cart mates directly to the test section. Water-cooled inserts are used in the test section to reduce its cross-sectional area. Upper and lower inserts in the termination section are used to smoothly transition the duct area to the full dimension at the exit. Conversion from the two- to the four-modular configuration and from the four- to the eight-modulator configuration is again accomplished through removal of the 2 x 4 and 4x8 transition carts, respectively. 922 HORN TTUNSTTION SECTION Figure 7: Two-modulator reduced test section configuration. Figure 8: Four-modulator reduced test section configuration. TEST PROCEDURE Measurements were taken for several conditions in each of the six facility configurations. Each modulator was supplied with air at a pressure of 207 kPa (mass flow rate of approximately 8.4 kg/s) and was electrically driven with the same broadband (40-500 Hz) signal. Acoustic pressures were measured at several locations along the length of the progressive wave tube using B&K model 4136 microphones and Kulite model MIC-190-HT pressure transducers, see Table 1. The positive x-direction is defined in the two-modulator full configuration (from the modulator exit) along the direction of the duct. The positive y-direction is taken vertically from the horizontal centerline of the 923 HORN TRIWSmOH SECTION MODULATOR TRANSITION FLEMSie HOSE adapter puts assembly TEST SECTION ADAPTER plate assembly TERM1ASAT10N SECTION horn TRANSITION CART Figure 9: Eight-modulator reduced test section configuration. duct and the positive z-direction is defined from the left sidewall of the duct as one looks downstream. Table 1: Kulite (K) and microphone (M) locations of acoustic measurements. Loc. Description Type Coordinate (m) 1 Test Sect. Horizontal Centerline Upstream K 7.75, 0, 0 2 Test Sect. Horizontal Centerline Downstream K 8.71, 0, 0 5 Test Sect. Vertical Centerline Top M 8,23, 0.3, 0 15 Test Sect. HorizontaWertical Centerline K 8,23,0, 0 25 Test Sect. Vertical Centerline Bottom M 8.23, -0.3, 0 28 2x4 HorizontaWertical Centerline M 2.19, 0,0 29 4x8 Horizontal Centerline, % Downstream M 3.66,0, 0 30 Horn Tran, Hor. Centerline, % Downstream M 4.75,0, 0 35 Termination HorizontaWertical Centerline M 12.46,0, 0.17 The acoustic pressure at location 1 was used as a reference measurement for shaping the input spectrum and for establishing the nominal overall sound pressure level for each test condition. For each configuration, the input spectrum to the air modulators was manually shaped through frequency equalization to produce a nearly flat spectrum at the reference pressure transducer. Data was acquired at the noise floor level (flow noise only) and at overall levels above the noise floor in 6 dB increments (as measured at the reference location) up to the maximum achievable. Thirty-two seconds of time data were collected at a sampling rate of 4096 samples/s for each transducer in each test condition. Post-processing of the time data was performed to generate averaged spectra and coherence functions with a 1-Hz frequency resolution. 924 RESULTS For each facility configuration, plots of the following quantities are presented: normalized input spectrum to the air modulators, minimum to maximum sound pressure levels at the reference location, maximum sound pressure levels in the test section, maximum sound pressure levels upstream and downstream of the test section, and vertical and horizontal coherence in the test section. The minimum levels in each case correspond to the background noise produced by the airflow through the modulators. Normalized input voltage spectra to each modulator for each configuration are shown in Figures 10, 15, 20, 25 and 30. These spectra were generated to achieve as flat an output spectrum as possible at the reference location for the frequency range of interest (40-200 Hz for the full section, 40-500 Hz for the reduced section). As expected, the significant difference between the full and reduced configurations is seen in the high (>200 Hz) frequency content. Figure 11 shows a background noise level of 126 dB (the lowest of all configurations) for the two-modulator full test section configuration. Nearly flat spectra are observed below 210 Hz for levels above 130 dB, giving a dynamic range of about 32 dB. The flat spectrum shape is a significant improvement over the performance of the old configuration as shown in Figure 2. Standing waves are evident at frequencies of 210, 340 and 480 Hz. For this reason, the full section operation is limited to less than 210 Hz or to the 220- 330 and 370-480 Hz frequency bands. The effect of standing waves are explored in further depth in the next section. The spectra in Figure 12 indicate a nearly uniform distribution in the x-direction throughout the test section. It is interesting to note that Figure 13 shows no sign of standing waves upstream of the test section, confirming that the cause is associated with the test section. Lastly, a near perfect coherence between upstream and downstream, and upper and lower test section locations is shown in Figure 14 for frequencies between 40 and 210 Hz. Again, this is a significant improvement over the performance of the old configuration (Figure 3). Figure 10: Normalized input spectrum Figure 11: Min to max SPL at location (2-modulator full). 1 (2-modulator full). 925 Figure 12: SPL in test section at max Figure 15: Normalized input spectrum level (2-modulator full). (4-modulator full). Figure 13: SPL along length of TAFA Figure 16: Min to max SPL at location (2-modulator full). 1 (4-modulator full). Figure 14: Test section coherence (2- Figure 17: SPL in test section at max modulator full). level (4-modulator full). The four-modulator full configuration exhibits similar behavior as the two- modulator full configuration as seen in Figures 16-19. The lowest level at which a uniform spectrum is achieved is 137 dB, giving a dynamic range of roughly 30 dB in this configuration. Lastly, the eight-modulator full 926 configuration results, shown in Figures 21-24, indicate a noise floor of about 142 dB and dynamic range of 22 dB. Frequency, Hz Frequency, Hz Figure 18: SPL along length of TAFA Figure 21: Min to max SPL at location (4-modulator full). 1 (8-modulator full). Frequency, Hz Frequency, Hz Figure 19: Test section coherence (4- Figure 22: SPL in test section at max modulator full). level (8-moduiator full). Frequency, Hz Frequency, Hz Figure 20: Normalized input spectrum Figure 23: SPL along length of TAFA (8-modulator full). (8-modulator full). 927 Frequency, Hz Frequency, Hz Figure 24: Test section coherence (8- Figure 27: SPL in test section at max modulator full). level (2-modulator reduced). Figure 25: Normalized input spectrum Figure 28: SPL along length of TAFA (2-modulator reduced). (2-modulator reduced). Figure 26: Min to max SPL at location Figure 29: Test section coherence (2- 1 (2-modulator reduced). modulator reduced). The reduced test section configurations are used to increase the frequency range and maximum sound pressure level in the test section. Results for the two-modulator reduced configuration, shown in Figures 26-29, indicate a nearly flat spectrum between 40 and 480 Hz, a noise floor of 129 dB and a dynamic range of about 28 dB. Coherence in the test section is nearly unity 928 over this frequency range. This represents a significant improvement over the old facility configuration. Results of similar quality indicate a d 5 mamic range of roughly 26 and 29 dB for the four- (Figures 31-34) and eight-modulator (Figures 36-39) configurations, respectively. Note that the coherence for these configurations is slightly reduced at the high frequencies, but is still very good out to 480 Hz. Figure 30: Normalized input spectrum Figure 33: SPL along length of TAFA (4-modulator reduced). (4-modulator reduced). 160 r 80 OASPLs: 134.1, 135.5, 142.1, 147.6, 153.9, 160.1, 165.9,167.9 1.0 ! 1 1 1 Loc 1, Loc 2 Loc 5, Loc 25 400 100 200 300 Frequency, Hz 400 500 Figure 31: Min to max SPL at location Figure 34: Test section coherence (4- 1 (4-modulator reduced). modulator reduced). Figure 32: SPL in test section at max Figure 35: Normalized input spectrum level (4-modulator reduced). (8-modulator reduced). 929 80 OASPLs; 134.1,135.7, 141.9,148.3, 154.1. 160.0, 165.9.170.5 80 Loc30 Loci 100 200 300 400 500 100 200 300 400 500 Frequency. Hz Frequency, Hz Figure 36: Min to max SPL at location Figure 38: SPL along length of TAFA i (8-modulator reduced). (8-modulator reduced). Figure 37: SPL in test section at max Figure 39: Test section coherence (8- level (8-modulator reduced). modulator reduced). Table 2 presents a summary of the maximum average OASPL for each facility configuration. In each case, the number of active modulators were run at maximum power as an independent group (independently for the single modulator case) and the results averaged. For example, results for one active modulator were obtained by running each modulator individually and averaging the resulting pressures. Table 2: Summary of maximum average overall sound pressure levels (dB). Number of Active Modulators 1 2 4 8 2-Modulator Red. 1 1 j 2-Modulator Full i 4-Modulator Red. 159.1 — MSM 4-Modulator Full 155.6 161.2 8-Modulator Red. 158.4 — mmm 171.7* 8-Modulator Full 153.0 158.4 164.5 170.0 ^Pressure scaled by ^7? from 7-modu ator run DISCUSSION In this section, limiting behaviors of the full and reduced test section configurations are explored and the effect of test section inserts, modulator coupling and wave tube performance are discussed. Limiting Behaviors The auto-spectra from the full test section configurations exhibit sharp reductions in level at approximately 210, 340, and 480 Hz. This behavior corresponds to measurements near nodes of vertical (height) standing waves in the test section portion of the wave tube. Table 3 summarizes theoretical, resonant frequencies and corresponding modal indices of the test section duct resonances within the excitation bandwidth. The modal indices m and n correspond to half wavelengths in the vertical and transverse (width) directions of the cross section, respectively. There are several resonances that may be excited below 500 Hz, but only three of these appear to be significant at the test section transducer locations (about the horizontal centerline). Because of the presence of air flow in the facility and lack of measurements in the cross section, it is difficult to correlate the experimental and theoretical modes. Measurements of the acoustic pressure at several locations in a cross-section of the duct will be necessary to fully characterize the resonant behavior. It is sufficient to say that the usable frequency range in the full test section configurations is approximately 40-210 Hz near the horizontal centerline. Acoustic pressure auto-spectra from the reduced test section configurations are essentially flat to almost 500 Hz. This is due to the fact that only two resonances are within the excitation bandwidth for this configuration, see Table 3. A sharp reduction is noted in the vicinity of 480 Hz. Although the (m=l, n=0) resonance does not appear to be significant, close inspection of the data (not shown) indicates its presence. Therefore, the usable frequency range for the reduced test section configurations is approximately 40-500 Hz. Table 3: Theoretical resonant frequencies of test section duct modes in Hz. Performance of Test Section Configurations For constant input acoustic power, the change from full to reduced test section configurations should theoretically result in a 4,7 dB increase in OASPL. However, Table 2 shows that increases of only 2.1 (e.g. 164.3-162.2), 0.9, and 1.7 dB were realized for the two-, four- and eight-modulator configurations. The system efficiency (actual/expected mean-square pressure) of the two-, four- and eight-modulator reduced configurations is 38, 40 and 44 percent, respectively, compared with 51, 63 and 62 percent for the two-, four- and eight-modulator full configurations. The expected pressure is calculated based upon a input-scaled value of the rated acoustic power of the WAS 3000 modulator assuming incoherent sources (3 dB per doubling). In general, the full section efficiency is greater than the corresponding reduced section efficiency. While the reason for this phenomena is not known, it is conjectured that the lack of expansion in the reduced configurations limits the development of plane waves. Therefore, phase and amplitude mismatches between acoustic sources may be accentuated. Modulator Coupling Performance A simplified waveguide analysis for coherent, phase-matched sources predicts increases in OASPL as shown in Table 4. Measured performance gains were less than predicted because of the assumptions of the waveguide analysis (inactive source area treated as hard wall), and possible reductions due to phase differences between modulators and non-parallel wave fironts at the exit of the modulator cart, see Figures 4-9. The latter effect is due to different angles of inclination of the sources relative to the axis of the wave tube. The greater gains achieved in the full test section configurations support the above contention that they are more efficient than the reduced configurations in combining the acoustic sources. Table 4: Change in SPL (dB) from 1 to max. number of active modulators. Configuration A SPL fi:om 1 Active Mod. (Meas/Pred) 2-Modulator Red. (2 active mods.) 3.9/6.53 2-Modulator Full (2 active mods.) 5.5/6.53 4-Modulator Red. (4 active mods.) 8.8 /13.98 4-Modulator Full (4 active mods.) 11.4/13.98 8-Moduiator Red. (8 active mods.) 13.3/22.10 8-Moduiator Full (8 active mods.) 17.0/22.10 Wave Tube Performance A change in configuration from the two- to the four-modulator configurations, and from the four- to the eight-modulator configurations, will result in an incremental increase of 3 dB in OASPL if the individual sources are phase- matched. This is due to a pure doubling of the power without any change in the radiation impedance of the individual sources. For the reduced configurations, a 3.6 and 3.8 dB increase are observed, respectively. A 4.8 and 932 3.0 dB increase are observed for the full configurations, respectively. Note that a greater than 3 dB increase is possible when the higher modulator configuration (for example, the four-modulator reduced configuration) is less susceptible than the lower modulator configuration (the two-modulator reduced configuration) to phase mismatches between modulators. This seems plausible because any such mismatches are averaged over a larger number of sources. SUMMARY Modifications to the NASA Langley TAFA facility resulted in significant improvements in the quality and magnitude of the acoustic excitation over the previous facility. The maximum OASPL was increased by over 6 dB (vs the previous 165 dB) with a nearly flat spectrum between 40-210 and 40-480 Hz for the full and reduced test section configurations, respectively. In addition, the coherence over the test section was excellent. These improvements, however, did not meet the objective for a maximum OASPL of 178 dB. There are several reasons why the maximum OASPL did not meet the objectives, including a lack of expansion in the reduced configurations and phase differences between modulators. A detailed computational analysis would be desirable to indicate the source of the inefficiencies and to help identify possible means of increasing the overall system performance. ACKNOWLEDGEMENTS The authors wish to thank Mr. H. Stanley Hogge and Mr. George A. Parker for their support in configuring and running the facility. We wish to also thank Mr. James D. Johnston, Jr. of NASA Johnson Space Center for loan of four Wyle air modulators. REFERENCES 1. Maestrello, L., Radiation from a Panel Response to a Supersonic Turbulent Boundary Layer, Journal of Sound and Vibration, 1969, 10(2), pp. 261-295. 2. Pozefsky, P., Blevins, R.D., and Langanelli, A.L., Thermal-Vibro- Acoustic Loads and Fatigue of Hypersonic Flight Vehicle Structure, AFWAL-TR-89-3014, 3. Williams, L.J., HSCT Research Gathers Speed, Aerospace America, April 1995, pp. 32-37. 4. Clevenson, S.A. and Daniels, E.F., Capabilities of the Thermal Acoustic Fatigue Apparatus, NASA TM 104106, February 1992. 933 SONIC FATIGUE CHARACTERISTICS OF HIGH TEMPERATURE MATERIALS AND STRUaURES FOR HYPERSONIC FLIGHT VEHICLE APPLICATIONS- Dr. I. Holehouse, Staff Specialist, Rohr Inc., Chula Vista, California 1. INTRODUCTION SUWiARY A combined analytical and experimental program was conducted to investigate thermal-acoustic loads, structural response, and fatigue characteristics of skin panels for a generic hypersonic flight vehicle. Aerothermal and aeroacoustic loads were analytically quantified by extrapolating existing data to high Mach number vehicle ascent trajectories. Finite-element thermal and sonic fatigue analyses were performed on critically affected skin panels. High temperature random fatigue shaker tests were performed on candidate material coupons and skin-stiffener joint subelements to determine their random-fatigue strength at high temperatures. These were followed by high temperature sonic fatigue tests of stiffened-skin panels in a progressive wave tube. The primary materials investigated were carbon-carbon and silicon-carbide refractory composites, titanium metal matrix composites and advanced titanium alloys. This paper reports on the experimental work and compares measured frequencies and acoustically induced response levels with analytically predicted values. The coupon shaker test data were used to generate material random fatigue "S-N" curves at temperatures up to 980°C. The joint subelements provided data to determine the effects on fatigue life of skin-stiffener joining methods. The PWT sonic fatigue panel tests generated response and fatigue life data on representative built-up skin panel design configurations at temperatures up to 925“C and sound pressure levels up to 165 dB. These^ data are used in determining the response strains and frequencies of skin panel designs when subjected to combined thermal-acoustic loading and to identify modes of failure and weaknesses in design details that affect_ sonic fatigue life. Sonic fatigue analyses of selected test panel design configurations using finite-element techniques were also performed and related to the experimental results. Acoustically induced random stresses were analytically determined on a mode-by-mode basis using finite element generated mode shapes and an analytical procedure that extends Miles approach to include multi-modal effects and the spatial characteristics of both the structural modes and the impinging sound field. The paper also describes the instrumentation development work performed in order to obtain reliable strain measurements at temperatures in excess of conventional strain gauge capabilities. This work focused primarily on the use of recently developed high temperature (350"C to 1000“C) strain gauges, laser Doppler vibrometers, high temperature capacitance displacement probes, and the determination of strain-displacement relationships to facilitate the use of double integrated accelerometer data to derive strain levels. 935 This work was funded by the USAF Flight Dynamics Laboratory (Kenneth R. Wentz, Project Engineer). The complete program report is contained in References 1 and 2. 2. HIGH TEMPERATURE STRAIN MEASUREMENTS Conventional adhesively bonded strain gauge installations are temperature limited to approximately 350°C. In order to achieve strain measurements at higher temperatures, up to QSO^C, ceramic layers and coatings were used to both attach strain gauges and to thermally protect them. However, such strain gauge installations are very sensitive to process parameters which often need varying depending upon the test specimen material. Coated carbon-carbon is a particularly difficult material to adhere to due to its material characteristics and relatively rough surface texture. Carbon- carbon also has a near zero coefficient of thermal expansion which presents attachment and fixturing problems in a high temperature environment. When high test temperatures either preclude or make problematic the use of strain gauges, an alternative technique for obtaining strain levels is to measure displacements and then determine strain levels using strain- displacement ratios. Strain is directly proportional to displacement for a given deflected shape, or mode shape, regardless of changes in the elastic modulus of the specimen material as it is heated. Consequently, if the deflected shape does not change significantly with temperature, high temperature test strain levels can be determined from room temperature strain and displacement measurements in combination with displacement measurements made at the test temperature. This measurement technique facilitates the use of non-contacting transducers which can be located away from the heated area, such as capacitance displacement probes or Laser Doppler Vibrometers (LDV). LDVs actually measure surface velocity but their signal outputs can be readily integrated and displayed as displacement. Accelerometers can also be used to measure displacement by double integrating their signal output. However, since accelerometers require surface contact they have to either withstand, or be protected from, the thermal environment. When this is not readily achievable, it is sometimes possible to install an accelerometer at a location on the test specimen or fixturing where the temperature is within its operating range, providing the displacement response at the point of measurement is fully coherent with the strain response at the required location. The displacement range limitations of the LDV and capacitance probes available to the program resulted in having to use double-integrated accelerometer outputs to measure displacements at room temperature and at the test temperature. Conventional strain gauges were used to measure strains at room temperature. In order to confirm that the strain- displacement ratios were unaffected by temperature, limited high temperature strains were measured at temperatures up to 980*C. Once the strain-displacement ratio for a given specimen type was determined, air¬ cooled accelerometers were used to determine high temperature test strain levels. The level of measurement accuracy of this technique was estimated to be within 10 percent. 936 The most successful strain measurements made at 980“C utilized a ceramic flame spray installation of an HFN type free filament gauge. This gauge installation included the use of silicon-carbide (SiC) cement as a base coat for the gage, applied over a 1-inch square area of a lightly sanded carbon-carbon surface substrate. Lead wire attachments to the gauge were made with standard Nichrome ribbon wire anchored to the specimen with SiC cement. With this gauge installation, it was possible to make dynamic strain measurements for short periods of time at 980“C. 3. RANDOM FATIGUE SHAKER TESTS The instrumented test specimens were mounted in a duckbill fixture and the specimen/fixture assembly then enclosed in a furnace. An opening in the furnace allows the specimen tip to protrude out in order to accorrmodate the air-cooled tip accelerometer. Figure 1 shows strain gauge locations and fixturing for material coupon and joint subelement specimens. The test procedure comprised a room temperature sine-sweep in order to identify the fundamental mode and its natural frequency, one-third octave random loading at room temperature centered around the fundamental natural frequency and one-third octave random endurance testing at the required test temperature and load level. Twelve inhibited carbon-carbon material coupons generated usable S-N data, eleven at 980“C and one at 650“C. S-N data points were also generated at 980°C for two integral joint and two mechanically fastened joint subelements. Fixturing problems and specimen availability limited the number of S-N data points generated. Figure 2 shows the random fatigue S-N data points with joint subelement data points superimposed. The random fatigue endurance level for the material coupons, extrapolated from 10 to 10® cycles, is approximately 320 microstrain rms. The integral joint subelements did not fail at the strain gauge locations; consequently, the actual maximum strain levels were higher than those shown on Figure 2. Taking this into account, it appears that the integral joints have a fatigue endurance level of greater than one-half of that for the material coupons. The mechanically fastened joint subelements exhibited fatigue strength comparable to that of the material coupons. These results indicate that carbon-carbon joints and attachments methods are not critically limiting factors in the structural applications of inhibited carbon-carbon. Figure 3 shows a representative example of the strain amplitude and peak strain amplitude probability density functions at room temperature for a material coupon specimen. The "peak" function can be seen to approximate a Rayleigh distribution, as it should for a Gaussian random process. Random fatigue S-N data were also generated for enhanced silicon-carbide composites (SiC/SiC) including thermally exposed specimens (160 hours at 980“C), titanium metal matrix composites (TMC) utilizing Ti 15-3 and Beta 21S titanium matrix materials, titanium aluminide (super alpha two), titanium 6-2-4-2, titanium 6-2-4-2-$i (including thermally exposed specimens) and Ti-1100. The fatigue endurance levels are shown in Table 1. Also shown in Table 1 are S-N data points for uninhibited carbon-carbon generated on a previous program (Reference 3). 937 TABLE 1. SUMMARY OF RANDOM FATIGUE ENDURANCE LEVELS. material TEMPERATURE ENDURANCE LEVEL CORRESPONDING TO 10® CYCLES: OVERALL RMS STRAIN (MICROSTRAIN) material COUPONS SUBELEMENTS INHIBITED CARBON-CARBON 1800“F (980"C) 320 integral JOINTS > 160 BOLTED JOINTS 320 5 PLY 1800T (980"C) 100 - UNINHlbl1tU CARBON-CARBON 1ft PI Y ISOO'-F igSOT) 150 _ *5 PLY 1000‘'F I540"C) 100 - 18 PLY lOOO'F (540-C) 450 - 5 PLY ROOM TEMPERATURE 550 - 18 PLY ROOM TEMPERATURE 450 - ENHANCED SiC/SiC NON-EXPOSED 1800"F r980“C) 450 - THERMALLY EXPOSED iaOO“F f980‘'C) 300 - 1000“F (540*0 520 DIFFUSION-BONDED HAT- STIFFENED = 520 n IML ROOM TEMPERATURE 2250 - beta 21S TMC 200 510 - TITANIUM ALUMINIDE /ciinCD ftl DUIA TUn^ ROOM TEMPERATURE 410 { rvui 1 iri • * Ti-6242-Si 735 Hilllll LIO BONDED HONEYCOMB BEAM = 388 ■■ WELDED JOINT = 400 T-; nnn - Ti 6-2-4-2 ROOM TEMPERATURE 675 - 866MISC/039-T1.IH 12-02-96 938 Figure 4 shows random fatigue S-N curves for the materials tested superimposed on one graph for comparison purposes. The Ti 15-3 TMC data are not shown since this was a concept demonstrator material utilizing a Ti 15-3 matrix material for producibility reasons. Ti 15-3 does not have the temperature capability for hypersonic vehicle applications. Titanium aluminide data are not shown due to its brittle material characteristics making it unsuitable for sonic fatigue design critical structures. Ti-1100 S-N data were very similar to the non-exposed Ti 6-2-4-2-Si and are not shown. Ti 6-2-4-2 coupons were only tested at room temperature before beingVeplaced by Ti 6-2-4-2-Si, which has higher structural temperature capabilities. The fatigue curves in Figure 4 show inhibited carbon-carbon to have higher fatigue strength at 980°C than does its uninhibited counterpart. Inhibited carbon-carbon also has greater resistance to oxidation at high temperatures. Although unexposed enhanced SiC/SiC had greater random fatigue strength at 980“C than did inhibited carbon-carbon, the two materials exhibited similar strength at temperature after allowing for thermal exposure. However, SiC/SiC has a maximum temperature capability of 1100 to 1200°C compared to 1700 to 1900“C for carbon-carbon. The Beta 21S TMC material demonstrated resonable fatigue strength at 815°C and the Ti 6-2-4-2-Si specimens exhibited high fatigue strength at 620°C to 650“C. 4. SONIC FATIGUE PANEL TESTS These tests were performed in Rohr's high temperature progressive-wave tube (PWT) test facility. The facility is capable of generating overal1^sound pressure levels of 165 to 168 dB at temperatures up to 925“C to 980“C, depending upon the test panel configuration and material. Three rib-stiffened carbon-carbon panels and a monolithic hat-stiffened Beta 21S TMC panel were subjected to sonic fatigue testing. Response strains were measured on the four panels over a range of incrementally increasing sound pressure levels (140 to 165 dB) at room temperature. One carbon-carbon panel was subjected to sonic fatigue testing at room temperature and the other two tested at 925“C. The TMC panel was endurance tested at 815“C. Figures 5 and 6 show a carbon-carbon panel and its fixturing installed in the PWT. The panels were attached to the fixture via flexures in order to allow for differences in the thermal expansion of the panel and fixture materials. Structural details of the panels and instrumentation locations are given in References 1 and 2. The three carbon-carbon panel configurations encompassed two skin thicknesses and two stiffener spacings as follows: Panel 1: 3 skin bays, 6 in. by 20 in. by 0.11 in. thick Panel 2: 2 skin bays, 9 in. by 20 in. by 0.11 in. thick Panel 3: 3 skin bays, 6 in. by 20 in. by 0.17 in. thick 939 Table 2 summarizes the measured room temperature frequencies and strain response levels: TABLE 2. ROOM TEMPERATURE RESPONSE OF TEST PANELS- TEST PANEL FREQUENCY OF IN-PHASE MODE (Hz) OASPL (dB) OVERALL RMS STRAINS rMICROSTRAIN) EDGE OF SKIN BAY CENTER OF SKIN BAY CARBON-CARBON NO. 1 267 165 305 149 155 & 171 145 126 39 191 59 165 558* 173* CARBON-CARBON NO. 3 423 165 69 127 BETA 21$ TMC 241 165 HIGHEST STRAIN = 287 AT PANEL CENTER ON STIFFENER CAP * EXTRAPOLATED ON THE BASIS OF TUE STRAIN RESPONSE WITH SPL FOR PANELS 1 AND 3. Panel 1 was subjected to 165 dB at room temperature for 10 hours at which point cracks developed at the ends of the stiffeners. The frequency dropped slightly during the ten hour test resulting in the number of cycles to failure being approximately 9 million. Panel 2 was endurance tested at 925“C at 150, 155 and 160 dB for 3-1/2 hours at each level, followed by one hour at 165 dB. At this point, cracks were observed at the ends of the stiffeners, similar to the cracks in Panel 1. Panel 3 was endurance tested at 925“C and 165 dB for 10 hours without any damage to the panel. The TMC panel was endurance tested at 815°C and 165 dB for 3-1/2 hours at which time cracks were observed in two stiffener caps at the panel center. The high test temperatures for Panels 2 and 3 and the TMC panel precluded attaching an accelerometer directly to the panel surface, even with air cooling. This prevented the direct measurement of panel displacements at 925°C. In order to attempt to estimate the high temperature endurance test strain levels, a temperature survey was performed on the panel fixturing with Panel 3 installed in order to determine an acceptable location for an accelerometer. An accelerometer at the selected fixture location tracked linearly with the highest reading strain gauges during a room temperature response survey. The coherence between the fixture accelerometer and the panel strain gauges was 0.9 in the frequency range of panel response. 940 Having established a coherent strain displacement relationship at room temperature, the temperature was increased progressively with increasing acoustic loading, generating accelerometer and microphone data at 480 C and 140 dB, 650°C and 155 dB, 860°C and 155 dB, and 925“C at 165 dB. It was clear from the data at the higher temperatures and load levels that the full spectrum overall rms displacement levels obtained by double integrat¬ ing the accelerometer output signals could not be used to determine high temperature strain levels due to high amplitude, low frequency displace¬ ments (displacement being inversely proportional to frequency squared for a given "g" level) that were well below the panel response frequency range and therefore would not be proportional to panel strain levels. It is important to remember here that since the accelerometer is mounted on the panel fixture, it is measuring fixture response, some of which is not related to panel response. After reviewing the various frequency spectra, it was decided to re-analyze the data to generate overall rms levels over selected frequency bandwidths that would encompass a high percentage of the full-spectrum overall rms strains and eliminate the low frequency displacements. If a consistent strain-displacement relationship could be established at room temperature within a frequency bandwidth such that the strains could be related to the full-spectrum overall rms strains, and if the same bandwidth could be used to generate displacements at temperatures that were sufficiently consistent to relate to strain response, then it would be possible to at least make a reasonable estimate of the test temperature strain level. It was determined that band-passed response data in the 300 to 600 Hz frequency range gave consistent strain-displacement ratios at room temperature. Double-integrated band-passed accelerometer outputs (displacements) were consistent with increasing sound pressure levels at incrementally increasing test temperatures up to the 925°C/165 dB endurance test conditions. Table 3 summarizes the high temperature test panel results. TABLE 3. HIGH TEMPERATURE TEST PANEL RESULTS. TEST PANEL TEST TEMPERATURE OVERALL SOUND PRESSURE LEVEL (dB) HIGHEST ESTIMATED OVERALL RMS STRAIN (MICROSTRAIN) EXPOSURE TIME, ESTIMATED FATIGUE CYCLES AND COMMENTS (°F) rc) BETA 21S TMC PANEL 1500 815 165 NOT ESTIMATED 3 1/2 HRS, 3x10“ CYCLES, STIFFENERS CRACKED AT MID-SPAN CARBON-CARBON PANEL NO. 2 150 155 TTTORSTTTMo^TYOlsr NO FAILURE 155 219 ^ 3 1/2 HRS, 2.3x10° CYCLtS, NO FAILURE 160 316 3 1/2 HRS. 2.3x10“ CYCLES, NO FAILURE 165 453 1 HR, 6.4xl0‘> cycles! CRACKS AT STIFFENER ENDS 925 165 103 10 HRS, 1.7x10' CYCLES, NO FAILURE 941 It should be noted that carbon-carbon panels 1 and 2 exhibited cracks at the stiffener ends, whereas the maximum measured strains were at the edges of skin bays. Consequently, the actual strain levels at the crack locations were either higher than the measured levels or there were significant stress concentrations at the stiffener terminations. 5. COMPARISON OF ANALYTICAL AND TEST RESULTS FOR CARBON-CARBON PANELS MSC NASTRAN was used to perform finite element analyses on the three carbon-carbon panels that were subjected to the sonic fatigue testing described in Section 4. The oxidation resistant coating was modeled as a non-structural mass, which is compatible with the panel test results.^ Natural frequencies, mode shapes and acoustically induced random strain levels were analytically determined for room-temperature conditions and compared to the room-temperature panel test results. Acoustically induced random stresses were analytically determined on a mode-by-mode basis using the finite element generated mode shapes and a Rohr computer code based on an analytical procedure presented in Reference 4. This procedure extends Miles* approach (Reference 5) to include multi-modal effects and the spatial characteristics of both the structural modes and the impinging sound field. Table 4 shows the calculated and measured frequencies, overall rms strain levels and the strain spectrum levels for the in-phase stiffener bending mode for the carbon-carbon panels at room temperature. TABLE 4. CALCULATED AND MEASURED RESPONSE FREQUENCIES AND STRAIN LEVELS FOR CARBON-CARBON PANELS AT ROOM TEMPERATURE. STRAIN LEVELS AT EDGE OF SKIN BAY NATUF FREQUE OF IN-F MODE (Hzl \fKL :ncy >HASE OVERALL RMS STRAIN (MICROSTRAIN) STRAIN SPECTRUM LEVEL IN-PHASE MODE (MICROSTRAIN/Hz) FE ANALYSIS MEASURED FE ANALYSIS MEASURED FE ANALYSIS MEASURED PANEL 1 a65 dBl 305 267 510 305 84 60 PANEL 2 (145 dBl 190 155 & 171 133 126 40 41 PANEL 3 (165 dB) 460 423 77 69 16 16 942 The above results show good agreement between the finite element generated values and those measured. The level of agreement is particularly good for the strain spectrum levels, which are typically more difficult to accurately predict. Figure 7 shows the finite-element frequency solution for Panel 3. The in-phase mode shape can be seen to have an overall modal characteristic due to the relatively low bending stiffness of the stiffeners for the skin thickness used. Figure 8 shows the measured and finite-element generated strain frequency spectra for Panel 3. Details of the finite-element analyses and models are contained in References 1 and 2. 6. CONCLUSIONS AND RECOMMENDATIONS 1. The high temperature testing techniques and strain measuring procedures successfully generated usable random fatigue S-N curves and panel response data. The use of strain-displacement ratios were shown to be an effective alternative to high temperature strain gauge measurements. 2. In general, the materials and structural concepts tested demonstrated their suitability for hypersonic flight vehicle skin panel applications. The major exception was Titanium-Aluminide Super Alpha Two which was determined to be too brittle. 3. Inhibited carbon-carbon exhibited significantly higher random fatigue strength at 980°C than did the uninhibited carbon-carbon — two to three times the random fatigue endurance strain level. 4. Thermally exposed enhanced SiC/SiC had comparable fatigue strength to that of inhibited carbon-carbon at 980°C. 5. The TMC specimens usefully demonstrated the fatigue strength of the TMC concept and the need to develop the concept to incorporate higher temperature capability titanium matrix materials. 6. Titanium 6-2-4-2-Si exhibited high fatigue strength in the 590°C to 650“C temperature range and also demonstrated the need for TMC materials to utilize higher temperature matrix materials in order to be cost effective against the newer titanium alloys. 7. The level of agreement between the finite element analysis results for the carbon-carbon panels and the progressive-wave tube test data demonstrated the effectiveness of the analytical procedure used. The analysis of structures utilizing materials such as carbon-carbon clearly presents no special difficulties providing the material properties can be well defined. 943 8 It is recommended that further tests be conducted similar to those performed in this program but with greater emphasis on testing panels having dimensional variations in order to develop design criteria and life prediction techniques. Such testing should be performed on those structural materials and design concepts that emerge as the major candidates for flight vehicle applications as materials development and manufacturing techniques progress. REFERENCES 1 R D. Blevins and I. Holehouse, "Thermo-Vibro Acoustic Loads and rkigue of Hypersonic Flight Vehicle Structure," Rohr, Inc. Engineering Report RHR 96-008, February 1996. 2. United States Air Force Systems Command, Flight Dynamics Laboratory Final Technical Report, Contract No. F33615-87-C-33^^/, to be published. 3. R. D. Blevins, "Fatigue Testing of Carbon-Carbon Acoustic Shaker Table Test Coupons," Rohr, Inc. Engineering Report RHR 91-087, September 1991. 4. R. D. Blevins, "An Approximate Method for Sonic Fatigue Analysis of Plates and Shells," Journal of Sound and Vibration, Vol. 129, 51-71, 1989. 5. J. W. Miles, "On Structural Fatigue Under Random Loading," Journal of Aeronautical Sciences, Vol. 21, November 1954. 944 A. Test Configuration for Material Coupons Fixture Specimen ■ O B. Test Configuration for Carbon-Carbon Integral Stiffener Specimens Accelerometer t C. Test Configuration For ■ Carbon-Carbon Mechanically Fastened Stiffener and All Titanium Diffusion Bonded Joint Specimens FIGURE 1 Typical Strain Gauge Locations and Test Configurations for Material Coupon and Joint Subelement Shaker Test Specimens 945 1200.0 o a CD CD '_< ji o o CD a CD o o O a o o CD CjD C\i j'lso JO nui ] unoJOS sw^ 1 i JSAO c o S- (/) ta ^ o c I OJ c E O QJ XI ■— i_ OJ rt3 xa C-5 3 CO -O I OJ +-» X> E •r— "r- xa o •r- f-Z) x: C J- *—• o 4- i- O t/a c- +-> c <u .f > o o. 3 cj cd -(-> z: ro I a . CO ( Z ( CU I ( 3 CO ( Oa ( -r- -O ' 4-> C cd cd 1 Ll_ ( CO ( E c ( o o < -o CL< C 3 cd O ac CJ CNJ LU or 3 o 946 Gfl=04 9 Fi?rau£?vCY= -4-^0 FIGURE? Finite Element Solution for In-Phase Overall Mode — Carbon-Carbon Panel, Concept 3 950 STRRIN [Ins xlB^G/Inl 952 SCALING CONCEPTS IN RANDOM ACOUSTIC FATIGUE BY Marty Ferman* and Howard Wolfe** ABSTRACT Concepts are given for scaling acoustic fatigue predictions for application to extreme environmental levels based on testing “ scaled” structures at existing, lesser environmental levels. This approach is based on scaling a test structure to fit within the capabilities of an existing test facility to attain fatigue results, and then using analytical extrapolation methods for predicting the full scale case to achieve accurate design results. The basic idea is to utilize an existing acoustic fatigue facility to test a structure which has been designed (scaled) to fatigue within that facility’s limits, employing the appropriate structural properties (such as thinning the skins, etc.). Then, the fatigue life of the actual structure is determined by analytically scaling the test results to apply to the full scale case ( thicker) at higher noise levels for example. Examples are given to illustrate the approach with limits suggested, and with the recognition that more work is needed to broaden the idea. BACKGROUND While it is important to continually expand the capability of acoustic test facilities , it is perhaps equally important to be able to work with existing facilities at any time. That is, facility expansions, enhancements , and modernization’s should always be sought from time to time, so long as practical and affordable from cost effective considerations. Limits should be pushed to accommodate larger sizes of test specimens with higher noise levels with wider ranges of frequencies, with wider ranges of temperatures, and with better capabilities for applying pressures along with any one of several types of preloads. These are costly considerations and require considerable time to accomplish. Facility rental can be used in some cases to bolster one’s testing facilities, however if the application suggests a situation beyond any available facility for the required design proof, then an alternate is needed. Thus the scaling concept suggested here is a viable and useable possibility. The Author’s basis for the approach stems from their extensive, collective, experience in Structural Dynamics, especially work in Acoustic Fatigue, Fluid-Structure Interaction, Buffet, and Aeroelasticity/Flutter, and particularly from experience with flutter model testing, in which it is quite common to ratio test results from a model size to full scale for valid Assoc. Prof, Aerospace and Mech. Engr. Dept, Parks College, St. Louis Univ., Cahokia, IL, 62206, USA ** Aerospace Engineer, Wright Laboratory, Wright Patterson AFB, OH, 45433, USA 953 predictions . Flutter is a well recognized area where model data is commonly used in nondimensional form to establish design margins of safety, as typified in Ref (1-2). Flutter can be nondimensionalized quite broadly as pointed out in many works, and is clearly done for a wide range of general cases using the “so-called” Simplified Flutter Concept, Ref (3). The degree of the use of the flutter model scaling rules varies considerably today, because some people are testing as much or more than ever, while others are testing less and relying more heavily on advanced theories such as Computational Fluid Dynamics, CFD. However, the basic ideas in flutter model scaling are still POWERFUL! In fact, this concept has fueled the Author’s desires to develop the “acoustical scaling” used in the approach presented. Moreover, when starting to write this paper, the Authors realized that this type of scaling is also common to many related areas of structural dynamics, and thus chose to include some examples of those areas to emphasize the main point here! For example, experience in fluid-structure interaction and fatigue of fuel tank skins, a related work area, serves as another example of scaling structures to demonstrate accurate predictions with widely varied environmental levels, and a multitude of configurations. Scaling and nondimensional results were used extensively in Ref (4 -13), and are cited here because of the immense data base accumulated. The work at that time did not necessarily define scaling as used here, but hindsight now suggests that there is a clear relation. It is becoming well recognized that Buffet is easily scaled , and many engineers and investigators are now employing scaling of pressures from model to full size applications, and are also using scaled model response to predict full scale cases . Some of the earliest and some of the more modern results clearly show this aspect. For example. Ref (14 -16) are typical, quite convincing, and pace setting regarding scaled data. Buffet models which are much more frail that the full-scale cases are used to develop data for full scale applications, and besides giving full sized results, provide a guide to safe flight testing as has been done more extensively with flutter testing. Obviously, acoustical response and fatigue phenomenon are also nondimensionalizable and scaleable. Ref (17-18), for example. This point is being taken further here; that is , scaling will be used to take better advantage of limited facility testing capability to predict more severe situations, as is used in the case of flutter model testing where a larger specimen is predicted from tests of a smaller structure using similarity rules. Here in the acoustic application, a thinner , or otherwise more responsive specimen, is tested and then analytical means are used to make the prediction for the nominal case. APPROACH The method is shown here is basically an extension of the flutter model scaling idea, as applied to acoustical fatigue testing with a particular emphasis on random applications. The technique will also work for sine type testing in 954 acoustical fatigue, and perhaps it will be even more accurate there, but most of today’s applications are with random testing, notably in the aircraft field. Thus it is in this area where the method should find more application . The Authors have a combined professional work experience of some 70+ years and thus have tried to focus this extensive background on an area where gains can be made to help reduce some costs while making successful designs, by using lesser testing capability than might be more ideally used. It is believed that the best testing for random acoustic fatigue, is of course, with (a) the most highly representative structure, and as large a piece as can be tested, both practically and economically, (b) the most representative environmental levels in both spectrum shape and frequency content, (c) test times to represent true or scaled time, as commonly accepted, (d) temperatures should be applied both statically and dynamically, and finally (e) preloading from pressures, vibration, and from boundary loading of adjacent structure. Frequently, testing is done to accomplish some goal using a portion of these factors, and the remainder is estimated . Thus the Authors believed that there is a high potential to extend the flutter model approach to acoustical applications. Recall that in the flutter model approach , the full scale flutter speed is predicted by the rule ((Vf)a)p- [ ((Vi.)m)e/ (( Vf)m)c] X [ ((Vf)a)c] (^) where Vp is flutter speed, the subscripts M and A refer to model and aircraft respectively, the subscript C refers to calculated, and the subscript P refers to predicted. Thus the equation suggests that the full scale predicted flutter speed is obtained by taking the ratio of experimental to calculated flutter speed for the model and then multiplying by a calculated speed for the airplane. These flutter model scaling ideas are covered in any number of References, i.e. Ref (lo), for example. The same concept can be utilized in acoustic fatigue, i.e. the strain at fatigue failure relation, (8,N) can be scaled from model structure tested at one level and then adjusted for structural sizing and environmental levels. This relation can be addressed as done for the flutter case: ((e,N)a)p=[(( s,N)m)e / ((e,N)m)c] X [((e.N)a)c] (2) where s is strain, and N is the number of cycles at failure, where as above in Eq (1), the subscripts M and A refer respectively to Model and Full Scale for parallelism, while the subscripts E, C, and P have the same connotation again, namely, experimental, calculated, and predicted. Thus the full scale case is 955 predicted from a subscale case by using the ratio of experimental to theoretical model results as adjusted by a full scale calculation. Flutter model scaling depends upon matching several nondimensional parameters to allow the scaling steps to be valid. While these same parameters are, of course, not necessarily valid for the acoustic relationships, other parameters unique to this acoustical application must be considered, and will be discussed. Accurate predictions for the method relies on extensive experience with the topic of Acoustic Fatigue in general, because concern is usually directed towards the thinner structure such as; panels, panels and stiffeners, and panels and frames, bays (a group of panels), or other sub-structure supporting the panels. These structures are difficult to predict and are quite sensitive to edge conditions, fastening methods, damping, combination of static and dynamic loading, and temperature effects. Panel response prediction is difficult, and the fatigue properties of the basic material in the presence of these complex loadings is difficult. However, the experienced Acoustic Fatigue Engineer is aware of the limits, and nonnally accounts for these concerns. Thus the method here will show that these same concerns can be accounted for with the scaling approach through careful considerations. The Authors believe that the method is best explained by reviewing the standard approach to acoustic fatigue, especially when facility limits are of major concern. Fig (1) was prepared to illustrate these points of that approach. Here it is seen that key panels for detail design are selected from a configuration where the combination of the largest, thinnest, and most severely loaded panels at the worst temperature extremes and exposure times are considered. These can be selected by many means ranging from empirical methods, computational means, and the various Government guides. Ref (17- 18), for example. Then detailed vibration studies are run using Finite elements , Rayleigh methods. Finite Difference methods, etc. to determine the modal frequencies and shapes, and frequently linearity is assessed. Then acoustical strain response of the structure is determined for sine, narrowband, and broadband random input to assess fatigue life based on environmental exposure times in an aircraft lifetime of usage. These theoretical studies are then followed by tests of the worst cases, where vibration tests are conducted to verify modal frequencies, shapes, and damping, and linearity is checked again for the principal modes. This is followed by acoustical strain response tests where the strain growth versus noise levels is checked, again employing sine, narrowband and broadband random excitation. Note the figure suggests that data from the vibration tests are fed back to the theoretical arena where measured data are used to update studies and to correlate with predictions, especially the effect of damping on response and fatigue, and of course, the representation of nonlinearity. Also, the measured strain response is again used to update fatigue predictions. These updates to theory are made before the fatigue tests are run to insure that nothing is missed. However, in this 956 case, the required sound presssure level SPL in (dB) is assumed to exceed the test chamber’s capability. Thus , as shown in the sketch in Fig (2) the key strain response curve, s vs dB, is extrapolated to the required dB level. This data is merged with the strain-to-failure curve at the right to establish the cycle count, N, giving the (s, N) point for this case. The extrapolated data provides some measure of the estimated life, but again is heavily dependent upon the accuracy of the basic strain response curve, and is especially dependent on whether high confidence exists at the higher strains. Linear theory is also shown in this case, indicating it overpredicts the test strain response and hence shows a shortened fatigue life compared to test data, as is generally the case in today’s extreme noise levels. This illustration is highly simplified, because experienced designers readily know that it is difficult to predict even simple panels accurately at all times, let alone complex and built-up structure consisting of bays (multi-panel); this will addressed again later in the paper. The new concept of scaled acoustic fatigue structures is shown on the sketch of Fig (3) where the standard method is again shown, but smaller in size, to refocus the thrust of the new idea. The scaling process parallels the standard approach, and actually complements that approach, so that the two can be run simultaneously to save time, costs, and manpower. Here the panel selection process recognizes that the design application requires environments far in excess of available facility capability. Thus the scaling is invoked in the beginning of the design cycle. As the nominal panels (bays) are selected and analyzed for vibration, response and fatigue, scaled structures are defined to provide better response within the existing chamber ranges so that they can be fatigued and then the results can be rescaled to the nominal case. In this manner, appropriate designs can be established to meet safety margins with more confidence, and will avoid costly redesign and retrofitting at downstream stages where added costs can occur and where down times are difficult to tolerate. The concept is further illustrated in the sketch of Fig (4). Here the strain response curve of the nominal case and that of the scaled version are combined with strain to failure data ( coupon tests) to show fatigue results. Note the strain response for the nominal case at the highest dB level available gives the fatigue value at point A, while the extrapolated data for this curve gives point B. The scaled model being more responsive gives the point C, and when rescaled gives the point D which differs slightly from the extrapolated point B as it most likely will, realistically. More faith should be placed on data from an actual fatigue point than a point based on the projected strain response curve. Note, Fig (5) illustrates the winning virtue of the scaleable design. The figure shows a hypothetical set of test data for the (s ,N) for a structure for various SPLs for the nominal case, open circles, and for the fatigue results of the scaled model , closed squares. The scaled model was assumed to be thinner here for example, and that the scaled data is also 957 rescaled to fit the nominal curve here. The most interesting aspect is shown by the two clusters of data, denoted as A and B where there are rough circles about drawn about the clusters. Here the emphasis is that tests of the scaled model ( and rescaled data) are used to find the higher strain conditions which cannot be found from the nominal case. In both cases at the highest strain levels, the facility is used to its limits, but with enough testing with the thinner case, adequate data is available to make the prediction more accurate using Eq (2) for the final correlation as shown here. The statistical scattering of the scaled data will be an accurate measure for the nominal case, particularly when compared to estimates based on extrapolation of the strain response for the nominal case. There are many cautions to be noted with this approach as there are with all acoustic fatigue methods, and of course, tests. First, the linearity of the modes, either in unimodal sine excitation, multi-mode sine, narrow band or broadband random must be carefully handled. The strain response of individual locations throughout the structure must be carefully monitored in calculations and tests so that strain response is truly understood and used to define fatigue life carefully. This is difficult to do in many applications where widely varying conditions and durations require some type of Miner Rule combination to provide a true measure of fatigue. Similarly, strain risers at fasteners, discontinuities, holes, frames, stiffeners, material changes along with temperature gradients, temperature transients, require final “tweeks” to predictions, regardless. Nonlinearity, especially in the multi -mode case, is one of the most formidable foes to conquer for any application. APPLICATIONS AND EXAMPLES The tests of an Aluminum panel of size 10x20 in. and with a thickness of 0.063 thickness , Ref (19), will be used to illustrate the technique. The panel has approximately fixed-fixed edge conditions, and is quite nonlinear in terms of strain response. Ref (19). The measured strain response for the panel is shown in Fig (6) along with an estimated response curve for a thinner panel ( 0.040 in ) based on the test data. For this case it is assumed that data were needed at 175 dB, while the facility could only achieve 164 dB. The strain response for the thinner case was estimated using the classical equation for the amplitude of response, 5 ; \\fdxdy\pSD,{f)] (3) where M is the generalized mass, co is the natural frequency, § is the viscous damping factor , (j) is mode shape, PSDp is the pressure Power Spectral Density, and x,y are the positional coordinates along the plate. Since strain , s, is proportional to the amplitude. 958 s= (t/2)(3(ti^/ax^)S (4) Combining Eq (3 and 4) shows that the strain response curves are proportional to the thickness factor, as given by: s 2 ~ ( ti / t2) (5) However, it must be noted that this case is nonlinear, and thus, this result is not exactly correct, but simply used for an illustration here. The actual data for the 0.063 thickness is extrapolated to a required 175 dB, showing a strain of 1000 micro in/in. The estimated curve for the thickness of 0.040 in. shows ,of course, a greater response at all dB levels as it should, and moreover shows that only 150 dB are needed to achieve the 1000 micro - strain condition. Moreover, the thinner panel will exhibit large enough strains at the lower SPLs to improve the fatigue curve where the thicker panel is insensitive. Taking the example a step further, the fatigue point of the nominal case is shov^Ti on a strain to failure plot in Fig (7), employing beam coupon tests of Ref (11), which were shown to be excellent correlators with panel fatigue in the collection of work in Ref (6-13). The fatigue point for the 164 dB excitation, 800 micro-strain, is shown as a triangle, while the extrapolated data for 175 dB is shown as the flagged triangle. One test point exists for the nominal case. Ref (20), and is shown by a star symbol. Data for the thinner panel are shown as circles at the various strain to cycle count cases for the various SPLs corresponding to the beam curve. Notably, these points can be seen to produce shorter fatigue cases as they should due to increased strains, but note that they are also at lower frequencies which would give a longer test time than if they were the nominal thickness. The scaled model is seen to produce the same point as the extrapolated case in this hypothetical case for the 1000 microstrain case ( again, at two different dB levels for two thicknesses). A SPL of 150 dB, rather benign, is seen to be quite effective. The actual fatigue point at 164 dB for the nominal case required 3 hours and was predicted to be 2.8 hrs. The estimated fatigue for the extrapolated case of 175 dB was estimated to be 1.7 hrs, while the scaled point from the thinner panel was estimated to be 2.2 hours which is slightly off, but the Authors have had to rely on log plots for much of the data and thus lack someaccuracy. Because of lack of actual data, the scatter from the estimate vs the test of the nominal case was used to scatter the estimate for the 1000 microstrain case, flagged dark circle, as if the use of Eq (2) had been employed directly. One must be careful here, because there can be a vast difference between theory and test, and this can mislead inexperienced persons applying these methods. As noted earlier, related work in fluid-structure and buffet , actually demonstrate this type of scaling. To illustrate, several figures are republished 959 here to make this point rather clear. Fig (8) of Ref (8 ) shows the dynamic bending strains in the bottom panel of an otherwise rigid fuel tank which is being excited vertically with moving base input. The vertical axis is strain while the abscissa is the number of g’s input. Three panel thicknesses and four depths of fluid (water in this case) were used. Note the sharp nonlinear effect in the response, rather than linear response growth as force increases. Interestingly, the data was nondimensionalized into the curve of Fig (9), Ref (8), which was originally intended for a design chart to aid in developing strain response characteristics for use in fatigue. This curve displays a parameter of response as the ordinate vs an excitation parameter on the abscissa. Here, E is Young’s modulus, p is density, t is thickness, a is the panel length of the short side, h is fluid depth, and the subscripts, p and F refer to panel and fluid, respectively. A point not realized previously is that the scaling shows that the thinner case can be used to represent the thicker panel under the appropriate conditions and when nonlinearity is carefully considered. More data with the thinner panels at the extreme conditions were unfortunately not taken in several cases of strain response because of concern with accumulating too many cycles before running the actual fatigue tests; else the thinner cases could have shown even more dramatically the scale effect. Buffet has been of more interest in the past 15 years because of high angle of attack operation of several modem USAF fighters. Much effort was placed upon research with accurately scaled models to detennine if these could be employed as in prior flutter work. The answer was YES! Several figures were taken from Ref (16) to illustrate scaling of data from a model of, a fraction of the size of a fighter, to the full scale quantity. Fig (10) shows the correlation between scaled-up model data, flight test, and two sets of calculations over a wide range of aircraft angle of attack for the F/A-18 stabilator. The data is for inboard bending and torsion moment coefficients produced by buffeting loads. The scaled model data correlates well, the calculations using Doublet Lattice (DLM) aerodynamics is close, while the strip theory is not as accurate. Fig (11) shows similar type of data for the F/A-18 Vertical Tail for outboard bending moment coefficients. Here a wider range of angle of attack was considered, and again scaled model data and calculations are close to aircraft values. Both cases suggest that model data can be used to supplement full scale work and that when combined with theory , are a powerful aid to full scale analysis and tests. These tests can be used early in the aircraft design cycle to insure full scale success. CONCLUSIONS AND RECOMMENDATIONS An attempt was made to employ a view of acoustical scaling different from that usually taken. The idea is to develop data for a model that fits within a test facility’s capability and then by using analytical methods, adjust these results to the nominal case using factors from the test based on the ratio of 960 experimental to calculated data. This is analogous to the flutter model approach. One example is offered, and similar results from related scaling in fluid -structure and buffet work were shown to further the point. While more work is needed to fully display the concept, enough has been done to inspire others to dig-in and more fully evaluate the approach. The Writers intend to do more research, since they fully appreciate this difficult task. REFERENCES 1. Scanlon , R.H., and Rosenbaum, R., “ Introduction to The Study of Aircraft Vibration and Flutter”, The MacMillian Company, New York, 1951 2. Bisplinghoff, R.L., Ashley , H. and , Halfman, R. L., “Aeroelasticity”, Addision-Wesley Publishing Co., NY, Nov. 1955, pp. 695-787 3. Ferman, M. A., “Conceptual Flutter Analysis Techniques - Final Report” Navy BuWeps Contract NO w 64-0298-c, McDonnell Report F322, 10 Feb. 1967 4. Ferman, M.A. and Unger, W. H., “Fluid-Structure Interaction Dynamics in Fuel Cells”, 17th Aerospace Sciences Meeting, New Orleans, La. Jan 1979 5. Ferman, M.A. and Unger, W. H. , “Fluid-Structure Interaction Dynamics in Aircraft Fuel Cells”, AlAA Journal of Aircraft, Dec. 1979 6. Ferman, M.A. , et al , “ Fuel Tank Durability with Fluid-Structure Interaction Dynamics ,” USAF AFWAL TR-83-3066, Sept. 1982 7. Ferman, M..A., Unger, W. H., Saff, C.R., and Richardson, M.D., “A New Approach to Durability Predictions For Fuel Tank Skins” , 26th SDM, Orlando, FL, 15-17 April 1985 8. Ferman, M. A. , Unger, W. H., Saff, C.R., and Richardson, M.D. , “ A New Approach to Durability Prediction For Fuel Tank Skins”, Journal of Aircraft, Vol 23, No. 5, May 1986 9. Saff, C.R., and Ferman, M.A, “Fatigue Life Analysis of Fuel Tank Skins Under Combined Loads”, ASTM Symposium of Fracture Mechanics , Charleston, SC, 21 March 1985 10. Ferman, M.A., Healey, M.D., Unger, W.H., and Richardson, M.D., “Durability Prediction of Parallel Fuel Tank Skins with Fluid-Structure Interaction Dynamics”, 27th SDM, San Antonio, TX, 19-21 May 1986 11. Ferman, M.A, and Healey, M.D., “Analysis of Fuel Tank Dynamics for Complex Configurations, AFWAL TR -87-3066, Wright-Patterson AFB, OH, Nov 1987 12. Ferman, M.A., Healey, M.D. and Richardson, M.D.,” Durability Prediction of Complex Panels With Fluid-Structure Interaction”, 29th SDM, Williamsburg, VA, 18-20 April 1988 13. Ferman, M.A., Healey, M.D., and Richardson, M.D., “A Dynamicisf s View of Fuel Tank Skin Durability, AGARD/NATO 68th SMP, Ottawa, Canada, 23-28 April 1989 14. Zimmerman, N.H. and Ferman, M.A., “Prediction of Tail Buffet Loads for Design Applications, USN Report, NADC 88043-30, July 1987 961 15. Zimmerman, N.H., Ferman, M.A., Yurkovich, R.N, “Prediction of Tail Buffet Loads For Design Applications”, 30th SDM, Mobil, AL , 3-5 April 1989 16. Ferman, M.A., Patel, S.R., Zimmemian, N.H., and Gerstemkom, G., “ A Unified Approach To Buffet Response Response of Fighters”,AGARD/NATO 70th SMP, Sorrento, Italy, 2-4 April 1990 17. Cote, M.J. et al, “Structural Design for Acoustic Fatigue”, USAF ASD- TR-63-820, Oct 1963 18. Rudder, F.F., and Plumblee, H.E., Sonic Fatigue Guide for Military Aircraft” AFFDL-TR-74-112, Wright-Patterson AFB, OH, May 1975 19. Jacobs, J.H., and Ferman, M.A. , Acoustic Fatigue Characteristics of Advanced Materials and Structures, “ AGARD/NATO SMP , Lillehammer, Norway, 4-6 May 1994 20. McDonnell Douglas Lab Report, Tech. Memo 253.4415, Acoustic Fatigue Tests of Four Aluminum Panels, Two With Polyurethene Sprayon”, 27 June 1984 Figure 1 - Standard Acoustic Fatigue Design Method 962 MICROSTRAIN - HYPOTHETICAL EXAMPLE SOUND PRESSURE LEVEL -dB N-CYCLES TO FAILURE Figure 2 - General Method for Estimating Fatigue Life at SPL above Test Facility Capability Figure 3 - Scaling Method Fits-in with General Design Cycle for Acoustic Fatigue 963 SOUND PRESSURE LEVEL - dB N@ cIBreq , Nom. Ext. N @ dBj^, Re-scaled N @ dBj^, Scaled Model N - CYCLES TO FAILURE Figure 4 - Hypothetical Examples of Acoustic Scaling to Tests at Higher SPLs HYPOTHETICAL CASE Figure 5 - Statistical Aspects of Scaling MICROSTRAIN - ^ MDC Tests Ref. (19) Figure 6 - Strain Response of an Aluminum Panel (10 X 20 X 0.063 in. 7075 T6) Narrowband Random Figure 7 - Example of scaled Model of the 10 x 20 in. Aluminum Panel 965 Symbol Panel Fluid Thickness Depth (in.) (in.) □ 0.032 11.0 0 0.040 11.0 o 0.063 11.0 0 0.032 8.0 A 0.040 8.0 o 0.063 8.0 0 0.032 4.0 0.040 4.0 • 0.063 4.0 Figure 8 - Dynamic Strain vs Excitation Level - Sine Figure 9 - Dynamic Strain Parameter vs Input Parameter Mean Strain at Fluid Depth 966 0.04 Inboard Bonding Inboard Torsion 0.032 RMS Buffet 0.024. Moment Coeffidant, 0.01 e 0.008 0 . 0 4 8 12 16 20 24 Angle of Attack - degrees A C 1 1 r 1 0 < ) • 0 _^ — Right t» Scaladt Calculal italculat — St ipmods 0 d(str^ ed(OUl Itast theory) ) — A L i > (j I o ( 0 ) 0 4 8 12 16 20 24 Angie of Attack - degrees Figure 10 - F/A-18 Stabilator Buffet Correlation Study 0.048 0.04 0.032 RMS Moment CoefficiantO.024 BM/(QL3) 0.016 0.008. 0 16 24 32 40 48 56 Angle of Attack - degrees Figure 11 - F-18 Vertical Tail Buffet Response Moment Coefficients for Angle-of-attack Variations Outboard Bending Moment (70% Span, 45% Chord) j—I j j — □ Ca OMC A FI — Liculatsd ideiscaX gniTastf — d4ip(MS 517 — WTTeet — 51. Run — 56) ( A 1 ( ' *, ) < ) " J h * ( > ( I 1 t 1 Zi 1 967 968 ACOUSTIC FATIGUE II THE DEVELOPMENT AND EVALUATION OF A NEW MULTIMODAL ACOUSTIC FATIGUE DAMAGE MODEL Howard R Wolfe WL/FIBG Bidg24C 2145 Fifth St Ste2 Astronautics Wright-Patterson AFB, OH 45433-7006 USA Robert G. White Head of Department Department of Aeronautics and University of Southampton Southampton, S017IBJ UK ABSTRACT A multimodal fatigue model has been developed for flat beams and plates. The model was compared with experimental bending resonant fatigue lifetimes under random loading. The method was accurate in predicting cantilevered beam fatigue lifetimes, but under predicted clamped- clamped beam test results. For the clamped plate tests, one calculation was accurate and the other predicted about half the test lifetimes. The comparisons and the parameters affecting them are presented. INTRODUCTION While the single mode acoustic fatigue theory is satisfactory for sound pressure levels around 158 dB overall and below, there is evidence in the literature [1, 2, and 3] that above this level the accuracy of the simple response prediction method decreases with increasing sound pressure levels. The purpose of the paper is two fold, first to develop a multimodal acoustic fatigue life prediction model ^d secondly, to evaluate its accuracy in estimating the fatigue life theoretically by comparing predictions with experimental results. FATIGUE MODEL DEVELOPMENT Many fatigue models are found in the literature. The Miner single mode model used by Bennouna and White [4] and Rudder and Plumblee [5] was selected to develop a multimodal nonlinear model. The fundamental formulation is given by, N.-fs^r «) 969 where N,. is the total number of cycles to failure, P(e) is the peak strain probability density. N is the total number of cycles to failure at incremental constant amplitude strain levels derived from a sinusoidal strain versus cycles to failure curve. To calculate the fatigue life in hour, Eq (1) can be expressed as, t (hours)= where t is time, Pp (sd) is the peak standard deviation probability density, Nc is the total number of cycles to failure at a specified strain level and ^ is the cyclic frequency. When the mean value is not zero, which is the case with axial strain in the beam or plate, the rms value is not the standard deviation. The standard deviation is usually employed to compute the time to failure. Most of the S-N curves or e -N curves are approximated as a straight line on a logarithmic graph. The relationship between the surface strain and the cycles to failure is then. 8 = (3) where K is a constant and a the slope of a straight line on a log-log graph. The cyclic frequency 4 for a single mode case is taken as the frequency of the associated resonance. Two types of peak probability density techniques were investigated from multimodal nonlinear strain responses [3]. These were called major peaks and minor peaks. The major peaks were counted for the largest peaks between zero crossings. The minor peaks were counted for all stress reversals or a positive slope in the time history followed by any negative slope. The effective cyclic frequency is much lower for the major peak count than the minor peak count. However, the peak probability density functions or PPDFs compared for these two cases were almost the same. The major peak method was selected for further study. Given a particular peak probability density curve from a measured response in an experiment, the number of peaks and the sampling time tg can be used to determine the effective multimodal cyclic frequency, fcm = number of peaks / tg (4) 970 where is the effective multimodal cyclic frequency. Substituting the multimodal cyclic frequency into Eq (4), t (hours) = Pp(sd) K/(e)‘/“ 3600xfcm (5) This model accounts for the effects of axial strains which cause the mean value not to be zero, nonlinear response and multimodal effects. If the mean value is zero, then the standard deviation is equal to the rms value and Eq (5) reduces to Eq (2). FATIGUE MODEL COMPARISON WITH EXPERIMENTAL RESULTS The peak probability density function (PPDF) is needed or preferably the time history from strain or dynamic response measurements to evaluate the fatigue model developed. Also needed are sinusoidal 8 -N curves for the structure, and knowledge of its boundary conditions and the equivalent multimodal cyclic frequency. Two types of peak probability density techniques were investigated from multimodal nonlinear strain responses [3]. These were called major peaks and minor peaks. The major peaks were counted for the largest peaks between zero crossings. The nunor peaks were counted for all stress reversals or a positive slope in the time history followed by any negative slope. The effective cyclic frequency is much lower for the major peak coimt than the nunor peak coimt However, the peak probability density functions or PPDFs compared for these two cases were almost the same. The major peak method was selected for further study. Comparison with Beam Data: The K and a terms were calculated from 8-N and S-N data, where S is stress, using Eq (3). Selecting two values of strain and their corresponding cycles to failure, yields two simultaneous equations which were solved for K and a. Two sinusoidal E-N curves for BS1470-NS3 aluminum alloy which has a relatively low tensile strength were obtained from Bennouna and White [4 Fig 8]. These were for a cantilevered beam and a clamped-clamped (C-C) beam as shown in Fig 1. The K and a terms calculated were used to compute the cycles to failure, Nc, for each strain level. Table I shown is the same as Table I in reference [4] except was calculated from Eq (3) to sum the damages. 971 Delta is the sample size. The cyclic frequency was for one mode the one resonant response frequency. The time to failure in hours using Eq (2) for the cantilevered beam was 16.6 hours compared with 16.2 predicted theoretically [4], 15.3 and 15.9 obtained experimentally [4]. Both theoretical results were essentially equal, but slightly higher than the test results. The time to failure from Eq (2) for the (C-C) beam was 3.04 hours compared with 2.53 theoretically [4] and 5.25 and 5.92 experimentdly [4]. Both theoretical results were about one half of the test results. The K and a terms, the theoretical fatigue life times and the experimental fatigue life times are listed in Table H. The table contains three sections: lifetimes calculated using a strain gauge PPDF, a displacement PPDF and the Gaussian and Rayleigh PDFs. As noted in reference [4], failure occurred much earlier for the C-C beam than the cantilevered beam for the same strain level. This was attributed to the influence of a large axial strain in the clamped- clamped beam. Comparison with Plate Data: Two fatigue tests were conducted to provide some additional limited data for comparison with the fatigue model developed. These tests used the base excitation method with a 1.09x10® N (20,000 Ibf) electrod)mamic shaker. The clamping fixture consisted of a flat aluminum alloy 6061-T6 plate 19 mm thick and four clamping bars of equal thickness. The radius of curvature of the clamping edges was 4.76 mm to prevent early fatigue failure. A four bar clamping arrangement was selected to prevent buckling of the plate while torquing the clamping bolts. The undamped size was 254 x 203 x 1.30 mm which results in a 1.25 aspect ratio. Strain gauges were bonded along the center of the larger dimension (SG 2) and at the center of the plate (SG 3). Displacements were measured with a scanning laser vibrometer at the center of the plate. An accelerometer was moimted on the shaker head to determine the acceleration imparted to the damped plate. A flat acceleration spectral density was used between 100-1500 Hz. Recordings were taken at increasing levels of exdtation up to the fatigue test level. The time to detecting the first fatigue crack was recorded for each plate. The constants K and a were calculated from random single mode S-N data for 7075-T6 aluminum alloy [5 p 489] shown in Fig 1, with K = 1.01x10^^ and a = - 0.175. The rms stress (Srms) ^^s changed to rms strain, Srms = EBrms, where E is Young's modulus. The stress was measured half way between two rivets along the center line between the rivets on the test specimen. The strain gauge location, stress concentrations, and the boundary conditions greatly affect the strain 972 level measured. Correction factors are needed for a different set of conditions and to convert random data to sinusoidal data. Sinusoidal £ -N bending coupon curves for 7075-T6 aluminum alloy were difficult to find. S-N curves were found for an aerospace material with both sinusoidal and random excitations. These curves were nearly parallel. The sinusoidal strain was 1.38 times larger than the random strain for 10^ cycles. Multiplying the constant K for the 7075-T6 material by 1.38 resulted in K = 1.40x10^^. Early strain gauge failures prevented strain measurement above 500 microstrain with 20.7 g rms shaker excitation. The fatigue test level was 115 g rms and the response contained at least six frequency response peaks. The major peak strain PPDFs were determined for 5.32 g rms and 20.7 g rms as shown in Fig 2 with the Gaussian PDF. Compared with the Gaussian distribution, an increased number of peaks occurred greater than 1 sigma and smaller than -1 sigma. Also a larger number of peaks occurred around zero. The PPDF determined from the 20.7g rms test case was used to predict fatigue life, but a new strain estimate was needed since the excitation level increased 5.6 times. The displacement is directly related to the strain at each excitation level. Displacement measurements at the fatigue test level were used to estimate the strain level shown in Fig 3. The estimated strain from the figure was 770 microstrain for SG 2. The scale of the displacement measurements was adjusted to coincide with strain measurements at increasing increments of shaker excitation. The equivalent cyclic multimodal frequency is needed to predict the fatigue life. Prediction of the linear modal frequencies is carried out by a variety of methods. Usually the first mode prediction is the most accurate. The cyclic multimodal nonlinear frequencies have been studied for two clamped beams and two clamped plates [3]. These were based upon the peak probability density functions (PPDFs) where the peaks were counted for a specific time interval, from which the nonlinear cyclic multimodal frequencies were calculated. Generally the resonant frequencies increased with increasing excitation levels. Those for the two beams increased more rapidly than those for the two plates. Very little change was noted for the plates. The equivalent cyclic miiltimodal frequency determined via Eq (4) from the SG 2 PPDF was 348 Hz. The time to failure in hours predicted using Eq (5) and the parameter mentioned above for the clamped plate was 0.706 hours compared with 1.17 and 0.92 shown in Table 11. The predicted result was slightly lower 973 than the test results. A Srms’^ curve was calculated with the test lifetimes available, by determining a new constant K, assuming the slope was the same as for the riveted coupon and applying the sinusoidal correction factor. The time to failure in hours predicted using Eq 5 and the calculated e-N curve for the clamped plate was 0.274 hours compared with 1.17 and 0.92 shown in Table E. The cyclic multimodal frequency used was the same at that determined from SG 2 and the same strain was used. This prediction was about 1/3 of the test results. This method incorporates the failure data at two points. The displacement PPDF shown in Fig 4 was used to predict fatigue life. The number of displacement peaks increased significantly above the strain PPDF around 1 sigma and -1 sigma. The large number of peaks around zero was similar to the characteristics observed in the strain PPDF. The equivalent cyclic multimodal frequency was 375 Hz, slightly higher than that determined from the strain PPDF. However, the same frequency (348 Hz) was used to predict fatigue life. The time to failure in hours using Eq (5) for the clamped plate was 1.15 hours with the riveted e-N curve and 0.446 with the calculated 8-N curve shown in Table E. This PPDF improved considerably the prediction. The Gaussian and Rayleigh PDFs were used to predict the time to failure with the same parameters as those used with the riveted e-N curve. The lifetime using the displacement PPDF was 1.15 hours, using the Gaussian PDF, 0.600 hours and using the Rayleigh PDF, 0.237 hours, as shown in Table E. The Gaussian PDF under predicts by a factor of 2. The Rayleigh PDF under predicts by a factor of 5. The spread sheets containing PPDF / Ncm data for various sigma values were used to determine damage accumulation shown in Fig 5. Almost 55% of the normalized damage occurs between -2 and -1 sigma and 38% of the damage between 1 and 2 sigma using the displacement PPDF. However, the damage is spread more evenly using the available strain gauge PPDF. The strain gauge PPDF was recorded at a much lower level than the displacement PPDF. The damage accumulation compared more closely to Gaussian PDF than the Rayleigh function. Damage Model with a Specific Function Describing the PPDF : A curve-fitting routine was used to determine a mathematical function for a high level strain gauge PPDF for the clamped shaker plate. The most important part of the fit is outside the range of -1 to 1 sigma, since most of the damage accumulation occurs outside this range. The highest ranking function was a tenth order polynomial followed by 974 ninth and eighth order polynomial fits. The goodness of fit in order from 1 to 14 ranges from 0.9775 to 0.9625, which are very close statistically. The tenth order polynonual is, y = a 4* bx + cx^ + dx^ + ex"^ + fx^ + gx^ + hx^ + ix^ + jx^ + kx^^ (6) where a=0.346, b=-0.0148, c=-0.137, d=-0.054, e=0.090, f=0.043, g=- 0.0400, h=-0.00976, i= 0.00722, j=6.85xl0"‘, k=<4.36xl0'" . Ranked fourteenth is a natural logarithmic function. The function and its coefficients are, Iny = a + bx + cx^ + dx^ +ex^ 4-fx^ (7) where a=-1.088, b=-0.1191, c=-0.1.302, d=0.0104, e=-0.0653, and f=0.0079. The function fits the test data similar to the tenth order polynomial and may be easier to use. Ranked forty-first is a Gaussian function. The function and its coefficients are. y = a + b exp|o.5[(x - c) / d]^ | (8) where a=-0.0968, b=0.4485, c=-0.050 and d=1.45. The function fits better for sigma values of 2 or greater than those of -2 sigma and greater. A constant coefficient is used to fit the Gaussian function to permit shifting the function to fit the test data. This equation can be used in the PPDF in Eq (5), a + b exp [-0.5[(x - c) / d]^ } - [K/(e)f“ 3600£ ■cm (9) where a=-0.0968, b=0.4485, c=-0.050 and d=1.45. CONCLUSIONS The prediction of multimodal fatigue life is primarily dependent upon the peak probability density function (PPDF) which changes shape with increasing excitation levels. The next in order of importance is the sinusoidal e -N bending fatigue curve and finally the effective multimodal cydic frequency. A multimodal fatigue model was developed with the PPDF estimated from a form of the Gaussian function being useful especially in the 975 range of cydes to failure from 10^-10^. The lifetime predication calailations for the damped-clamed beam was about one half the experimental value. For the plate, the calculations was about one half the experimental value. Using riveted coupon fatigue data, the calculation was accurate. REFERENCES 1. B.L. Clarkson, April 1994, '‘Review of sonic fatigue technolog/', NASA contractor report 4587, NASA Langley Research Center, Hampton, Virginia. 2. R.G. White, October 1978, "A comparison of some statistical properties of the responses of aluminium alloy and CFRP plates to acoustic excitation". Composites 9(4), 125-258. 3. H.F. Wolfe, October 1995, "An experimental investigation of nonlinear behaviour of beams and plates excited to high levels of dynamic response", PhD Thesis, University of Southampton. 4. M. M. Bennouna, and R. G. White, 1984, "The effects of large vibration amplitudes on the dynamic strain response of a clamped- clamped beam with consideration on fatigue life". Journal of Sound and Vibration, 96 (3), 281-308. 5. J. R. Ballentine, F. F. Rudder, J. T. Mathis and H.E. Plumblee, 1968, 'Refinement of sonic fatigue structural design criteria", AFFDL TR 67-156, AD831118, Wright-Patterson AFB, Ohio. TABLE I FATIGUE CALCULATIONS USING EQUATION 5.9 CANTILEVERED BEAM BS 1470-NS3 (REF 4 FIG 8) e/sd sd P£ e ll£ Nc=(2.172 xl0“/sd)“““ PPDF X A PPDF X A/Nc t hours 0.5 H 213 15079045.41 n 1.41256E-07 1 |g 425 4390936.857 mam 7.06000E-07 1.5 425 638 2125446.064 WBrn 1.12917E-06 2 425 850 1273261.536 0.138 1.08383E-06 2.5 425 1063 854026.6252 0.060 7.02554E-07 3 425 1275 617189.6449 0.018 2.91645E-07 3.5 TOTAL 425 1488 468372.0662 0.005 0.984 1.06753E-07 4.16121E-06 16.7 976 TABLE n SUMMARY OF FATIGUE CALCULATIONS Figure 1 8 -N curves for aluminum alloys tested. Figure 2 Normalized strain PPDF comparison with a Gaussian PDF. 978 NORMALIZED DISPLACEMENT MOJORPPDF 2 Figure 4 Normalized displacement PPDF comparisons, 9 980 ACOUSTIC FATIGUE AND DAMPING TECHNOLOGY IN COMPOSITE MATERIALS By B. Benchekchou and R.G. White Abstract Considerable interest is being shown in the use of composite materials in aerospace structures. Important areas include development of a stiff, lightweight composite material with a highly damped, high temperature polymer matrix material. The study described in this paper concerns the application of such material in the form used in thin skin panels of aircraft and investigation of its fatigue properties at room and high temperature. For this purpose, flexural fatigue tests have been carried out at two different temperatures and harmonic three-dimensional FE analyses were performed in order to understand the dynamic behaviour of plates. Random acoustic excitation tests using a progressive wave tube, up to an overall sound pressure level of 162 dB, at room temperature and high temperatures were also performed in order to investigate the dynamic behaviour of panels made of the materials. Various methods for including damping in the structure were examined when parameter studies were carried out, and conclusions have been drawn concerning optimal incorporation of a highly damped matrix material into a high performance structure. 1-Introduction Significant areas of primary and secondary structures in military aircraft operate at high temperature and are subjected to high levels of random acoustic loading, because of their closeness to jet effluxes. There is then a need to develop a carbon fibre reinforced plastic material with a high temperature polymer matrix and high fatigue resistance. Highly damped composite structures should be developed in order to better resist dynamic loading and to have an enhanced fatigue life. Work previously carried out on improving the damping in fibre reinforced plastic (FRP) composites as well as the number of approaches which can be taken to improve the damping properties of polymeric composites have been summarised in [1]. The aim of the research described here was to study lightweight composite materials with a highly damped, high temperature polymer matrix material, by 981 investigating its mechanical and acoustic fatigue properties, the latter investigation being carried out using thin, multilayered plates. 2-ExperimentaI work For this type of study, two adequate prepregs were highlighted after investigation: SE300 and PMR15. The SE300 material was carbon fibre reinforced prepreg of (0°/90°) woven form, 0.25 mm thick and had 60% fibre volume fraction, with no suitable data available on the material properties. Dynamic mechanical thermal analyser (DMTA) analyses carried out on four specimens ( 20 nun long and 12 mm wide) with different lay-ups i.e. (0°/90°)4, (+45°/-45°)4, (0°/90°, 45°/45°)s and (45®/45°;0®/90°)s, allowed to get provided the material properties. Results from DMTA analyses are shown in Table l.a where the loss factor and the Young’s modulus values at 40^C and at the glass transition temperature Tg are presented. The loss factor values varied from 0.0097 to 0.085 for a range of temperature from 40®C to 300®C. The PMR15 prepreg was also of (0^/90^) woven carbon form and had 58% volume fraction. Six DMTA specimens having the following lay-ups: (0®/90®)4, (00/90^)8, ( 4450 /- 450 ) 4 , (+450/-45<^)8, ( 0 ^/ 90 ^-, 450/450)s and ( 450 / 450 . 00 / 900 )^ were made. Results from DMTA analyses showed that the loss factor values varied from 0.0129 to 0.0857 for a range of temperature from 40®C to 400®C, with a value of loss factor of 0.1293 at 375®C, the maximum temperature for normal use being 352^C. The loss factor and Young’s modulus values at 40®C and Tg are given in Table l.b. Mechanical behaviour of the selected materials The fatigue characteristics of these new materials were investigated and results were compared with those of well established structural materials. Mechanical fatigue tests of SE300 and PMR15 samples using "sinusoidal" loading at a chosen maximum strain level, i.e. 8000 pS were carried out and performances compared to that of an XAS/914 sample. A mechanical (flexural) fatigue rig was used for this purpose to test specimens in a cantilevered configuration. Details of the rig are available in [2]. The particular clamp used was designed by Drew [3] to induce damage in the centre of the specimen instead of having edge damage, i.e. peeling while flexural tests are carried out. In order to investigate the performance of these new materials at high temperature, fatigue tests were also carried out on samples, at 210^C . This was achieved by using a heating system which consisted of two air 982 blowers (electronically regulated hot-air guns) positioned at 40 mm above and below the specimens, which allowed specimens to be tested at a uniform temperature of 210±5°C. The aim of the mechanical fatigue test was to determine the number of loading cycles needed for damage to occur and its subsequent growth rate in cyclically loaded composite specimens of SE300 and PMR15 matrix materials . The samples were 140 mm long, 70 mm wide and 2 mm thick. Fatigue tests of SE300 (S3 and S4) and PMR15 (PI and P2) specimens, at room temperature and at 210OC respectively, at a level of 8000 p.S, located by the peak of the half-sine clamp, have been carried out. Ultrasonic scans of specimens S3 and S4 before any loading cycles and after 100, 500, 1000, 2000, 5000, 10000, 20000 and 50000 loading cycles are shown respectively in Figures l.a-h and Figures 2.a-h. A small delamination, indicated by lighter areas in the scans, starts to show in both specimens S3 and S4 after applying 500 loading cycles and increases substantially after 5000 loading cycles. After 5000 loading cycles, the damage area increased more for specimen S4 than specimen S3, which shows that the latter is slightly more fatigue resistant. In other words, when increasing the temperature from 250C to 210<^C, the resistance to fatigue slightly decreases. Figures 3.a-h and Figures 4.a-h show the ultrasonic scans for PMR15 specimens PI and P2 before and after several loading cycles. For both specimens PI and P2, damage starts after 500 loading cycles and increases substantially after 20000 loading cycles. At this stage, delamination areas are similar for both specimens PI and P2 and just a little more pronounced in specimen P2, which shows that the latter is slightly less fatigue resistant. Hence, an increase in temperature leads to a decrease in the fatigue resistance properties of the specimens. From Figures 1, 2, 3 and 4, one can conclude that the PMR15 specimens tested were slightly more fatigue resistant than the SE300 specimens. In fact, damage in specimens was generally more defined, clearer and spread more rapidly in the SE300 samples than was the case for the PMR15 samples. Figure 5 shows damage propagation occuring in an XAS/914 sample (XI), with (0o/±45®/0O)s stacking sequence, tested at 8000 |iS level and at room temperature, from [3]. Note that substantial damage existed after 1000 loading cycles in this specimen, which shows that both SE300 and PMR15 are more fatigue resistant than XAS/914 at room temperature. Acoustic fatigue behaviour of panels of the selected materials Investigations were carried out by installing the CFRP plates in an acoustic progressive wave tunnel, (APWT) in order to determine the response of CFRP plates under broadband acoustic excitation simulating jet noise. The plate was fully clamped around its boundaries on to a vertical steel frame fixed to one side of the APWT, so that it formed one of the vertical walls of the test section of the APWT. 983 Overall sound pressure levels (OSPL) up to 165 dB of broadband noise in the test section of the tunnel was generated by a Wyle Laboratories WAS 3000 siren. A heater panel capable of heating and maintaining the temperature of test plates up to 300*^0 while mounted on the tunnel was designed and built. Temperatures were monitored and controlled via thermocouples on the panel. Plates were excited by broadband excitation in the frequency range 80-800 Hz. A B&K type 4136 microphone mounted at the centre of the test section of the tunnel adjacent to the mid-point of the plate were used for sound pressure measurements. Eight strain gauges, four on each side of the plate were attached in order to monitor the strain distribution in the panel while the rig was running; more details of the experiment may be found in [1]. Acoustic tests were run, at various temperatures and OSPL. Since plates were excited in the frequency range 80-800 Hz, spectral analyses would not include the first natural frequency. The natural fundamental frequency of an SE300 clamped panel was found to be 49.02 Hz analytically. The second -and third resonance frequencies were 149.5 and 198.5 Hz, as calculated from strain spectral densities, from tests carried out with an OSPL of 156 dB and a temperature of 1620C. At 162 dB, results showed that the second resonance frequency was 113 Hz at 150OC and 106.5 Hz at 195^0, which shows that when the temperature and the OSPL increased, the resonance frequencies of the plates decreased. Also, it was found that the damping increased at elevated temperatures. The overall modal viscous damping ratios, for the second mode, were calculated from strain spectral densities, for an SE300 panel driven at an OSPL of 162 dB and at 1950C, and was found to be 8.91%; this value is similar to that calculated from analytical simulations, for the first mode, which is 8,50% at 242^0, (see the analytical section below). A typical strain spectral density obtained from recorded results is presented in Figure 6 for an SE300 specimen, at an OSPL = 156 dB and at 1620C, from a strain gauge in the centre of the specimen. Maximum RMS strain values recorded from experimental tests, at a strain gauge in the centre of the specimen were, at an OPSL =156 dB, 1300 \iS, 1800 \iS and 2800 pS at room temperature and at 90^0 and 1620C respectively. These results clearly indicate a trend for significant increase in dynamic response with increasing temperature. Experiments were also carried out on a PMR15 panel at various OSPL and temperatures. Results from tests run at 159 dB and at room temperature show that the second and third resonance frequencies were indicated as 112 and 182.5 Hz. When the OSPL increased to 162 dB, the second and third resonance frequencies decreased to 110.5 Hz and 176.5 Hz. At the same OSPL (162dB) and when temperature increased to 2810C, the third resonance frequency became 139 Hz. This 984 shows that, for PMR15 plates, when the temperature and the OSPL increased, the resonance frequencies of the plates decreased. It was also clear that modal damping increased with increasing temperature. In fact, the overall viscous damping reached 20%, for the second mode, at an OSPL=162 dB and at 2810C. It must be stated here, however, that apparent damping trends could include nonlinear effects which influence bandwidths of resonances. Maximum RMS strain values recorded at room temperature, by a strain gauge in the centre of the plate were found to be: 2700 pS at 153 dB, 2800 pS at 157.9 dB and 2900 pS at 159 dB. It is clear from these values that increasing the OSPL obviously leads to an increase of the strain in the plate. Similar results were observed when the temperature was increased. In fact, at an OSPL of 162 dB, the maximum strain values recorded by a strain gauge in the centre of the specimen were 3000 pS, 3400 pS and 5000 pS at 105^0, 1650C and 281^0 respectively, which clearly indicates the effects of temperature. It was observed that both the PMR15 and SE300 panels behaved in a non linear manner. Attempts to acoustically fatigue a PMR15 panel were made at 162 dB. No signs of fatigue damage were shown in an ultrasonic scan of the panel after 1389 minutes of running time. 3-Analytical work In order to examine various methods for including damping in a structure, parametric studies were carried out using the finite element FE method. ANSYS software has been used. A three-dimensional, 3D layered element, SOLID46 was used to build theoretical models. The element is defined by eight nodal points, average layer thickness, layer material direction angles and orthotropic material properties, [4]. Meshes were built in order to carry out modal and harmonic analyses of multilayered composite plates (410 mm, 280 mm, 2 mm). The plates were fully clamped along all edges, in order to simulate the panels tested in the APWT. Natural frequencies were first determined from free vibration analyses and compared to resonance frequency values derived from experimental data. Then, the plate was driven by harmonic loading at one point of application. The forcing frequency varied from 0 to 400 Hz. The amplitude of the load was 50 N. Results for displacements and response phase angles relative to the force for a chosen position on the plate as a function of frequency were obtained. The approach was then to carry out parameter studies in order to examine various methods for including damping in the structure, i.e. to use highly damped matrix material throughout the whole structure or possible 985 incorporation in a few layers. Structural damping was included, allowing models to run with different damping values in each ply of the panel. Structural damping is inherent in the structure and depends on the natural frequency; details on structural damping modeling may be found in [1]. Analyses were performed considering structural damping for the first mode. The structural damping was then varied for plies with the same orientation for a viscous damping ratio ^ = 0.01,0.02, 0.05, 0.10 and 0.20. Simulations with SE300 Models were built up with the following stacking sequence ((45°/45°),(0°/90°))s, lay-up used for the experimental plates. Table 2 gives the first three modal frequencies of the panel obtained from free vibration analyses results. Harmonic simulations were carried out and the overall damping value was calculated for each case with results given in Table 3. As can be seen, if high overall damping is needed for a structure composed of the SE300 material, increasing the damping value of the (45°/45®) orientation plies most significantly increases the overall damping value of the panel. In fact, putting a damping value of 20% in the (45°/45°) orientation plies leads to an overall viscous damping value of 14.52%, which is better than including a 10% damping value in all of the plies of the structure. Harmonic analyses of fully clamped plates were also performed with the values of material properties taken at several temperatures. Simulations were carried out with material properties at 2420C and 300OC. Free vibration analyses permitted calculation of the modal frequencies of the panels at the temperatures mentioned above. Table 2 also lists the first three modal frequencies from analyses with material properties at 242^0 and 300^0. The overall viscous damping values, obtained from FE simulations, are given for each ternperamre in Table 4. Again, the damping value has been varied through the layers and the overall damping value was calculated in order to see which of the plies contributes the most to heavily damp the plate. It was found that putting a damping value of 20% in the (45°/45°) orientation plies, the first mode viscous damping ratios were 14.62% and 14.55% at 242^0 and 300OC respectively. This shows that this material is more highly damped at high temperature and presents better damping properties of the two materials at 242^0. Simulations with PMR15 Free vibration analyses of models built up with the following stacking sequence ((45°/45°),(0°/90®))s were carried out and the first three modal frequencies of the panel are shown in Table 5. Harmonic analyses were run and the overall damping 986 value was calculated for each simulation with results given in Table 6. If high overall damping is needed for a structure composed of the PMR15 material, increasing the damping value of the (45°/45®) orientation plies most significantly increases the overall damping value of the panel. In fact, putting a damping value of 20% in the (45°/45°) orientation plies leads to an overall viscous damping value of 14.39%, while if (0°/90®) orientation plies have a 20% damping value, the overall damping is 7.42%. Harmonic analyses of fully clamped plates were also carried out with the values of material properties taken at several temperatures. Simulations were carried out using material properties at 3750C and 400oC. Free vibration analyses permitted calculation of the modal frequencies of the panels at the temperatures mentioned above. Table 5 lists the first three modal frequencies from analyses with material properties at 3750C and 400^0. The overall viscous damping values, obtained from FE simulations, are given for each temperature in Table 7. Again, the damping value has been varied through the layers and the overall damping value was calculated in order to see which of the plies contributes the most to heavily damp the plate. It was found that putting a damping value of 20% in the (45‘^/45°) orientation plies, the first mode viscous damping ratios were 18.39% and 16.94% at 3750 c and 400OC respectively. This shows that this material is more highly damped at high temperature and presents better damping properties of the two materials at 3750C. 4~Conclusions Two matrix materials, SE300 and PMR15, with potential for use in aircraft structures in a severe environment, i.e. temperatures up to SOO^C were selected for this study. Material properties were determined using DMTA techniques and results show that these materials have high damping abilities at high temperature. Dynamic loading tests, performed in flexure at room and high temperature showed that the carbon fibre reinforced PMR15 material is more fatigue resistant than SE300 and XAS/914 based composites. Acoustic tests using a progressive wave tunnel, up to a random acoustic OSPL of 162 dB, at room temperature and elevated temperamres up to 2810C were also performed. When increasing the excitation level and the temperature higher strain values in the centre of the panels were recorded. Free vibration and harmonic FE analyses permitted determination of the natural frequencies and the overall viscous damping values. Resonance frequencies determined from results obtained from acoustic tests were similar to 987 natural frequencies obtained from FE simulations. Overall viscous damping values obtained from experimental results agreed well with those obtained from the FE analyses for SE300 panels. Results obtained for PMR15 panels, from tests, were higher than those calculated analytically. Both tests and simulations showed that SE300 and PMR15 present higher damping capabilies at high temperatures. Conclusions, via parameter studies including material damping, have been drawn concerning optimal incorporation of a highly damped matrix material into a high performance structure. 5-AcknowIegments The authors wish to thank the Minister of Defence for sponsorship of the programme of research under which the work was carried out. Thanks are also due to Dr M. Nash of the DRA, Famborough for many helpful discussions throughout the project. _ 6-References 1- Benchekchou, B. and White, R.G., Acoustic fatigue and damping technology in FRP composites, submitted to Composite Structures. 2- Benchekchou, B. and White, R.G., Stresses around fasteners in composite structures in flexure and effects on fatigue damage initiation: I-Cheese-head bolts. Composite structures, 33(2), pp. 95-108, November 1995. 3- Drew, R.C. and White, R.G., An experimental investigation into damage propagation and its effects upon dynamic properties in CFRP composite material . Proceedings of the Fourth International Conference on Composite Structures, Paisley College of Technology, July 1987. 4- ANSYS theoretical manual, Swanson Analysis Systems Inc, December 1992. 988 Table La: Loss factor and Young’s modulus values at 40^C and at Tg for SE300 samples analysed by the DMTA. stacking sequences (4.45/-45)4 (-457+45)4 (45/45;0/90)s (0/90)4 Tg(°C) 242 242 240.3 238.71 T| atTg 0.085 0.085 0.081 0.061 Ti at 40°C 0.012 0.014 0.010 0.0097 Log E’ at40°C 9.870 9.840 9.970 10:097 Table Lb: Loss factor and Young’s modulus values at 40^C and at Tg for PMR15 samples analysed by the DMTA. Stacking sequences (+45/--45)4 (0/90;45/45)s (45/45;0/90)s (0/90)4 Tg (OC) 372 375 375 375 Tl atTg 0.117 0.124 0.121 0.129 T| at 40®C 0.0110 0.0138 0.0086 0.0117 Log E’ at 40OC 9.583 9.944 9.972 9.875 989 Table 2: The first three modal frequencies for SE300 panel; analyses carried out with material propert ies at room temperature, at 242*^0 and at 300°C. Room temperature 242^0 300OC 49.02Hz 44.72Hz 43.61Hz 155.80Hz 143.27Hz 140.17Hz 212.04Hz 194.19Hz 189.68Hz Table 3: Overall viscous damping values of SE300 panel. Values are calculated from results obtained from harmonic analyses; the material damping being considered for the first mode. Simulation with damping of (45°/45°)orientation plies (0®/90°)orientation plies 5% 3.81% 2.51% 10% 7.06% 3.92% 20% 14.52% 11.57% Table 4: Overall viscous damping values of SE300 panel, for the first mode. Temperature (°C) Overall viscous damping 25 1.20% 242 8.50% 300 5.45% 990 Table 5: The first three modal frequencies for PMR15 panel; FE analyses carried out with material properties at room temperature, 375°C and 400°C. Room temperature 3750c 400OC 43.43Hz 32.68Hz 25.54Hz 130.35Hz 99.65Hz 78.83Hz 183.96Hz 139.04Hz 109.11Hz Table 6; Overall viscous damping values of PMR15 panel. Values are calculated from results obtained from harmonic analyses; the material damping being considered for the first mode.. Simulation with damping of (45°/450)orientation plies (0o/90O)orientation plies 5% 3.76% 2.44% 10% 7.26% 4.10% 20% 14.39% 7.42% Table 7: Overall viscous damping values of PMR15 panel, for the first mode.. Temperature Overall viscous damping (OQ 25 1.33% 275 13.24% 400 8.6% 991 a: before any loading cycles b: after 100 loading cycles c: after 500 loading cycles d: after 1000 loading cycles e: after 2000 loading cycles f: after 5000 loading cycles g: after 10000 loading cycles h: after 20000 loading cycles Figure 1. Ultrasonic scans of specimen S3 after applying different numbers of loading cycles. (SE 300 material, ambient temperature) a: before any loading cycles b: after 100 loading cycles c: after 500 loading cycles d: after 1000 loading cycles e: after 2000 loading cycles f: after 5000 loading cycles Fisure 2. Ultrasonic scans of specimen S4 after applying different numbers of loading cycles. (SE300 material, 210OC) 992 a: before any loading cycles b: after 100 loading cycles c: after 500 loading cycles d: after 2000 loading cycles e: after 20000 loading cycles f: after 50000 loading cycles g: after 100000 loading cycles Figure 3. Ultrasonic scans of specimen PI after applying different numbers of loading cycles (PMR15 material, ambient temperature, 8000 ]LlS) a: before any loading cycles b: after 500 loading cycles c: after 1000 loading cycles g: after 10000 loading cycles h: after 20000 loading cycles Figure 4. Ultrasonic scans of specimen P2 after applying different numbers of loading cycles. (PMR15 material, 2100C, 8000M.S) 993 a: before any b: after 100 c: after 500 d: after 1000 loading cycles loading cycles loading cycles loading cycles e: after 2000 f: after 5000 g: after 10000 h: after 20000 loading cycles loading cycles loading cycles loading cycles Figure 5. Ultrasonic scans of an XAS/914 specimen fatigued at a level of 8000 llS showing the damage propagation; the lay-up is (0/±45/90)s, [3], 0 Lin Hz RCLD 1.6k Figure 6: SE300 specimen SI strain spectral density, recorded from strain gauge ST2, OSPL =156 dB, temperature = 162^C. 994 THE BEHAVIOUR OF LIGHT WEIGHT HONEYCOMB SANDWICH PANELS UNDER ACOUSTIC LOADING David Millar Senior Stress Engineer Short Bros. PLC Airport Road Belfast Northern Ireland SUMMARY This paper discusses the results of a progressive wave tube test on a carbon composite honeycomb sandwich panel. A comparison was made with the test panel failure and the failure of panels of similar construction used in the intake ducts of jet engine nacelles. The measured panel response is compared with traditional analytical methods and finite element techniques. Nomenclature = Overall rms stress (psi) or strain (jxs). 7t =3.14159 = Fundamental frequency (Hz). 5 = Critical damping ratio (*0.017). Lps(fn) = Spectrum level of acoustic pressure (- expressed as a fluctuating rms pressure in psi in a 1 Hz band). jr = Joint acceptance function (non dimensional). = Characteristic modal pressure (psi) ph = Mass per unit area (Ib/in^) Sic “ Modal stress (psi) or modal strain (|j£). Wjc = Modal displacement (in), a = Panel length (in), b = Panel width (in). x,y,z = Co-ordinate axes. 1.0 Introduction Honeycomb sandwich panels have been used for some time in the aircraft industry as structural members which offer a high bending stiffness relative to their weight. In particular, they have proved very attractive in the construction of jet engine nacelle intake ducts where, in addition to their load carrying ability, they have been used for noise attenuation. 995 2.0 Acoustic Fatigue The intake duct of a jet engine nacelle can experience a severe acoustic environment and as such the integrity of the nacelle must be assessed with regard to acoustic fatigue [1], Acoustic fatigue characterises the behaviour of structures subject to acoustic loading, in which the fluctuating sound pressure levels can lead to a fatigue failure of the structure. The traditional approach to acoustic fatigue analysis has assumed fundamental mode response and given that aircraft panels will in general, have fundamental frequencies of the order of several hundred hertz, it is clear that the potential to accumulate several thousand fatigue cycles per flight can exist. Techniques for analysing the response of structures to acoustic loads were developed originally by Miles [2] and Powell [3], Other significant contributions are listed in References 4-7. Design guides such as AGARD [8] and the Engineering Sciences Data Unit (ESDU) series of data sheets on vibration and acoustic fatigue [9], have proved useful in the early stages of design. Note - further details on the general subject of acoustic fatigue can be found in Ref 10, while a more detailed review of the subject up to more recent times is presented in Ref 11. 3.0 In Service Failures In recent years a number of failures have been experienced involving intake barrel honeycomb sandwich panels. Failures have been experienced with panels which had both aluminium facing and backing skins and carbon composite panels. The metal intake liner was observed to have skin cracking and also core failure, while the composite panel was only observed to have core failure. With regard to the metal panels, flight testing was carried out and the predominant response frequency was observed to be at the fan blade passing frequency - much higher than the fundamental frequency of the intake barrel; this went some way to explaining why the traditional approach in estimating the response did not indicate a cause for concern. The response of the panel was also very narrow band - almost a pure sinusoid (again differing from the traditional approach of broad band/random load and response), and the subsequent analysis of the results was based on a mechanical fatigue approach [12]. Subsequent fleet inspections revealed that core failure was observed prior to skin failure and it was assumed that the skin failure was in fact caused by a breakdown in the sandwich panel construction. The core was replaced with a higher density variety, with higher shear strength and moduli. This modification has been in service for several years with no reported failures. The modification represented only a moderate weight increase of the panel, without recourse to changing skin thickness, which would have proved very expensive and resulted in a substantial weight penalty. As mentioned above, another intake duct, of carbon composite construction, also began to suffer from core failure. The panels of this duct had a carbon backing skin while the facing skin had a wire mesh bonded to an open weave carbon sheet. The only similarity was the use of the same density of honeycomb core (although of different cell size and depth). For other reasons this core had been replaced by a heavier variety, prior to the discovery of the core failures and the impact of the failures was minimised. 996 Limited data is available on similar failures and only 2 other cases, regarding nacelle intake barrels, appear to have been documented [13 & 14], however neither case involved sandwich panels. A number of theories had been put forward as to the cause of the failures. These included neighbouring cells resonating out of phase, cell walls resonating or possibly the panel vibrating as a 2 degree of freedom system (the facing and backing skins acting as the masses, with the core as the spring) - this phenomenon had originally been investigated by Mead [15]. 4.0 Physical Testing A number of tests were carried out with "beam" type high cycle fatigue specimens and also small segments of intake barrel. None of these tests were able to reproduce the failures observed in service (Figure 1.0) which further served to reinforce the belief that the failures were attributed to an acoustic mechanism as opposed to a mechanical vibration mechanism, however in an attempt to cover all aspects it was decided to carry out a progressive wave tube (PWT) test on an abbreviated panel. For simplicity it was decided to test a flat sandwich panel of overall dimensions 36"x21" (Figure 2.0). The panel was instrumented with 12 strain gauges and 2 accelerometers. Two pressure transducers were also mounted in the fixture surrounding the specimen. Testing was carried out by the Consultancy Service at the Institute of Sound and Vibration Research (IS VR) at the University of Southampton. 4.1 PWT Results The panel was first subject to a sine sweep from 50 to 1000 Hz in order to identify its resonant frequencies. The response of a strain gauge at the centre of the panel has been included in Figure 3.0. On completion of the sine sweeps, the linearity tests were carried out.. As only 8 channels could be accommodated at one time, it had been decided to arrange the parameters into 5 groups, with each group containing 4 strain gauges, 2 accelerometers, 1 pressure transducer on the fixture and 1 pressure transducer in the PWT (this was required by the facility for the feedback loop). The initial tests were carried out with a power spectral density of the applied loading constant over the 100 Hz to 500 Hz range, however when using this bandwidth only 155 dB overall, could be achieved. In an attempt to increase the strain levels it was decided to reduce the bandwidth to 200 Hz. The bandwidth (BW) was subsequently reduced to 100 Hz and finally 1/3 octave centred on the predominant response frequency of the panel. When failure occurred a dramatic change in response was observed. The failure mechanism was that of core failure as shown in Figure 4.0. There was no indication of facing or backing skin distress. 997 5.0 Comparison With Theoretical Predictions 5.1 Fundamental Frequency From the strain gauge readings the panel was seen to be vibrating with simply supported edge conditions. Soovere [7] suggests that "effective" dimensions (essentially from the start of the pan down) be used to determine the fundamental frequency which is given by; x\n This equation is applicable to simply supported panels with isotropic facing and backing skins, thus for the purpose of applying the above equation, the actual section was approximated to a symmetric (isotropic) section. The predicted fundamental frequency is given below. It was observed however, that if the panel dimensions are taken relative to mid way between the staggered pitch of the fasteners a significant improvement was achieved (see "Soovere (2)" in table 1). Alternative frequency estimations using an FE model and an ESDU data item [16] are summarised in the following table; Method Freq.(Hz) % Error Measured 228 - Soovere 274.3 +20.3 Soovere (2) 213.3 -6.4 FE 239.04 +4.8 ESDU 193 -15.3 Table 1 - Comparison of calculated frequencies for simply supported sandwich panel. Note; the percentage error is based on the actual measured response frequency of the panel in the PWT. Given that the excitation bandwidth extended (at least initially) up to 500 Hz, modes up to 500 Hz were obtained from the FE model. In actual fact 2 FE models were used, the first was a basic model with 380 elements, however a more detailed model, shown in Figure 5.0, (with essentially each element split into 4) was used for the results presented in this paper. The predicted modes from the FE model were as follows; Mode No. Frequency (Hz) Mode No. in x direction (m) Mode No. in y direction (n) Figure No. 1 239.04 1 1 6 2 334.0 2 1 7 3 430.02 1 2 8 4 1 9 Table 2 - Finite Element Model Predicted Frequencies. 998 5.2 rms Strain The predicted strains were calculated using Blevins' normal mode method (NMM) [5], with a joint acceptance of unity for the fundamental mode of vibration, using the following expression; s^= Lp.(fJ . ^ (2) In an attempt to improve the estimated response, the rms strain was calculated for each mode within the bandwidth of excitation. The Joint accetptance for each mode was calculated using equation 3 and the calculated strains for each mode were then factored by the relevant joint acceltance term. The overall strain was then calculated for all the relevant modes. A comparison with ESDI! [16] has also been included, however the ESDU method does not provide an indication of shear stress in the core. Soovere presents a simple expression for the joint acceptance function for a simply supported panel excited by an (acoustic) progressive wave, for the case where n is odd; •2 _ ^ (l-Cos(m7c)Cos(c0ra/c) , . (1 - (cD^a/mrcc)^) Note, when n is even the joint acceptance is zero. Given that the bandwidth varied for the applied loading, the overall SPLs were expressed as spectrum levels for the purpose of comparison in the linearity results, the results (both measured and predicted) have been summarised in table 3, (SGI results have been plotted in Figure 10.0). The results from the ESDU data item [16] have been included in table 4 for comparison. Note - due to recorder channel limitations SGI & SG2 were not connected at the time of failure and no results were available at the highest sound pressure levels. OASPL (dB) Spectrum Level SPL (dB) Measured Strains(u£) SGI SG2 Calculated (ps) j=l SGI SG2 Calculated (Multi Mode) SGI (us) SG2 130 107 7 7 8.7 9.2 2.5 2.6 140 117 20 19 27.5 29.1 8.0 8.1 150 127 55 60 87.0 92.0 25.4 25.8 155 132 100 100 154.7 163.7 45.1 45.8 157 134 130 130 194.8 206.0 56.8 57.6 163 140 - - 388.7 411.1 202.5 162.7 164 141 - - 436.1 461.2 312.0 250.7 Table 3 - Comparison of Measured & Predicted rms Strains for the Panel Centre, Facing & Backing Skin Gauges. 999 OASPL (dB) Spectrum Level SPL (dB) Measured Strains (pe) SGI SG2 ESDI! Strains SGI SG2 130 107 7 7 10.1 29 140 117 20 19 150 127 55 60 101 290 155 132 100 179.5 515.7 157 134 130 ■E9 253.6 728.5 163 140 - - 637 1830 164 141 - - 1010 2900 Table 4 - Comparison of Measured & ESDI! Predictions of the rms Strains for the Panel Centre, Facing & Backing Skin Gauges. There is a considerable difference in the calculated response from using a joint acceptance of unity for the fundamental mode and that when estimating the joint acceptance for each mode and calculating the overall response for several modes, however it was observed that if the average value from both methods is used the response compares favourably with that measured (-at least for the cases under consideration). The average value has been included on the linearity plot for SGI, shown in Figure 10.0).In general, the level of agreement between theory and practice was considered adequate and it was decided to apply the theory to estimating the shear stresses in the core (Table 5); OASPL (dB) Spectrum Level SPL (dB) Core Shear Stress (J=l) (rms psi) Core Shear Stress (Multi Mode) (rms psi) Average Core Shear Stress (rms psi) Peak Core Shear Stress _(E£!i_ 130 107 0.3 0.09 0.19 0.58 140 117 0.94 0.28 1.84 150 127 2.99 0.88 1.93 5.8 155 5.31 1.57 3.44 10.32 157 134 6.69 1.97 4.33 13.0 163 140 13.35 5.57 9.46 28.38 164 141 14.98 8.59 11.78 35.34 Table 5 - Predicted Core Shear Stress. 6.0 Discussion & Recommendations The ESDI! method proved very conservative and will thus give a degree of confidence when used in the early stages of the design process. Blevins Normal Mode Method was observed to give reasonable accuracy in predicting the highest strains in the panel and would merit use when designs have been fixed to some degree; at which stage FE models become available. For panels whose predominant response is in the fundamental mode it is accepted that the contribution from shear to overall deformation is very small. The main concern when designing a honeycomb sandwich panel which is subject to "severe" acoustic 1000 loads has tended to focus on skin strains and to some degree the properties of the core material have been ignored. The fact that low skin strains are observed has the effect of giving an impression that there is no cause for concern, however when the properties of the core material are low or unknown, some caution is required. There is unfortunately no available S-N data for the type of honeycomb used in the construction of the panel, however the allowable ultimate strength for the core material is of the order of 26 psi, so clearly the 163 dB level was sufficient to cause a static failure while the lower SPLs can be assumed to the have contributed to initiating fatigue damage. On cutting up the test panel, a large disbond was observed however it did not extend to the panel edge where cracking had occurred (the mid point of the long edge being the location of maximum shear for a simply supported panel) and it was the opinion of the materials department that the failure had not initiated in the disbond. The SPLs used in the test were not excessively high and were comparable to service environments (an example of which is given in Table 6). It should be noted that while the levels in Table 5 are 1/3 octave bandwidths, the actual spectrum is not generally flat within each band for engine intakes, but is rather made up of tones (Figure 11). These tones or spectrum levels can thus essentially be the band level and thus some caution should be exercised when converting intake band levels to spectrum levels using the traditional approach [17]. 1/3 Octave Centre Sound Pressure Frequency (Hz) Level (dB) 100 141 125 133 160 140 200 142 250 140 315 139 Table 6 - Typical Acoustic Service Environment. Note; Overall levels may reach 160 - 170 dB, however they tend to be influenced by SPLs at blade passing frequencies, which are much higher than panel fundamental frequencies. 7.0 Conclusion It has been shown that although moderate levels of acoustic excitation produce quite low overall rms strains in the skins of honeycomb sandwich panels, it is still possible, when using very light weight cores, to generate core shear stresses of a similar order of magnitude to the allowable ultimate strength of the material. Acknowledgements The author acknowledges the support of Short Bros. PLC in the course of preparing this paper and also the assistance of Mr Neil McWilliam with regard to the FE modelling. 1001 References I. 0 Air worthiness Requirements (JAR/FAR) Section 25.571 .d. 2.0 Miles, J.W., "On Stmctural Fatigue Under Random Loading," Journal of the Aeronautical Sciences, (1954),Vol.21, p753 - 762. 3.0 Powell, A., "On the Fatigue Failure of Structures due to Vibrations Excited by Random Pressure Fields,” Journal of the Acoustical Society of America, (1958), Vol.30, No.l2,pll30- 1135. 4.0 Clarkson, B.L., "Stresses in Skin Panels Subjected to Random Acoustic Loading," Journal of the Royal Aeronautical Society, (1968), Vol.72, plOOO- 1010. 5.0 Blevins, R.D., "An Approximate Method for Sonic Fatigue Analysis of Plates & Shells," Journal of Sound & Vibration, (1989), Vol.129, No.l, p51-71. 6.0 Holehouse, I., "Sonic Fatigue Design Techniques for Advanced Composite Aircraft Structures," AFWAL TR 80-3019,(1980). 7.0 Soovere, J., "Random Vibration Analysis of Stiffened Honeycomb Panels with Beveled Edges," Journal of Aircraft, (1986), Vol.23, No.6, p537-544. 8.0 Acoustic Fatigue Design Data (Part 1), AGARD-AG-162-72, (1972). 9.0 ESDU International, London, Series on Vibration & Acoustic Fatigue. 10.0 Richards, E.J., Mead, D.J.,"Noise and Acoustic Fatigue in Aeronautics," John Wiley & Sons, New York, (1968). II. 0 Clarkson, B.L., "A Review of Sonic Fatigue Technology," NASA CR 4587, (1995). 12.0 Millar, D., "Analysis of a Honeycomb Sandwich Panel Failure," M.Sc. Thesis, University of Sheffield, (1995). 13.0 Holehouse, L, "Sonic Fatigue of Aircraft Structures due to Jet Engine Fan Noise," Journal of Sound & Vibration, (1971), Vol. 17, No.3, p287-298. 14.0 Soovere, J., "Correlation of Sonic Fatigue Failures in Large Fan Engine Ducts with Simplified Theory," AGAEUD CPI 13 (Symposium on Acoustic Fatigue), (1972), pi 1-1 - 11-13. 15.0 Mead, D.J., "Bond Stresses in a Randomly Vibrating Sandwich Plate: Single Mode Theoiy," Journal of Sound & Vibration, (1964), Vol.l, No.3, p258-269. 16.0 ESDU Data Item 86024 (ESDUpac A8624), "Estimation of RMS Strain in Laminated Face Plates of Simply Supported Sandwich Panels Subjected to Random Acoustic Loading," Vol. 3 of Vibration & Acoustic Fatigue Series. 17.0 ESDU Data Item 66016, "Bandwidth Correction," Vol. 1 of Vibration & Acoustic Fatigue Series. 1002 Backing Skin Figure 2.0 - PWT Test Specimen. 1003 Microstrain (dB) Figure 3.0 - Response of Strain Gauge SGI During Sine Sweep. Figure 4.0 - Section Through Failure Region in PWT Panel. 1004 Figure 5.0 - PWT Panel Finite Element Model. Figure 8.0 - FE Mode 1 (m==l, n=2) Figure 9.0 - FE Mode 2 (m=3, n=l) 1005 Sound Pressure Level (dB) Figure 10.0 - Linearity Plot for SGI (Measured & NMM Prediction). Figure 11.0 - Typical Spectral Content of Intake Duct Sound Pressures with Equivalent 1/3 Octave Levels Superimposed. 1006 Time Domain Dynamic Finite Element Modelling in Acoustic Fatigue Design Authors: P. D. Green Military Aircraft British Aerospace Warton A. Killey Sowerby Research Centre British Aerospace Filton Summary Advanced Aircraft are expected to fly in increasingly severe and varied acoustic environments. Improvements are needed in the methods used to design aircraft against acoustic fatigue. Since fatigue life depends strongly on the magnitude of the cyclic stress and the mean stress, it is important to be able to the predict the dynamic stress response of an aircraft to random acoustic loading as accurately as possible. The established method of determining fatigue life relies on linear vibration theory and assumes that the acoustic pressure is fully spatially correlated across the whole structure. The technique becomes increasingly unsatisfactory when geometric non-linearities start to occur at lugh noise levels and/or when the structure is significantly curved. Also the excitation is generally not in phase across the whole structure because of complex aerodynamic effects. Recent advances in finite element modelling, combined with the general availability of extremely fast supercomputers, have made it practical to carry out non-linear random vibration response predictions using time stepping finite element (FE) codes. Using the time domain Monte Carlo (TDMC) technique it is possible to model multi-modal vibrations of stiffened aircraft panels without making the simplifying assumptions concerning the linearity of the response and the characteristics of the noise excitation. The technique has been developed initially using a simple flat plate model. This paper presents some of the results obtained during the course of this work. Also described are the results of a study of the “snap-through” behaviour of the flat plate, using time domain finite element analysis. For simplicity, it was assumed that the dynamic loading was fully in phase across the plate. Introduction Aircraft structures basically consist of thin, generally curved, plates attached to a supporting framework. During flight these stiffen^ panels are subjected to a combination of static and dynamic aerodynamic loads. On some aircraft there may be additional quasi-static thermal loads due to the impingement of jet effluxes in some areas. Parts of advanced short take of and landing (ASTOVL) .aircraft may be required to withstand noise levels up to 175dB and temperatures up to 200deg C. Under these conditions the established methods of dynamic stress analysis for acoustic fatigue design are inappropriate and cannot be employed. British Aerospace (BAe), Sowerby Research Centre (SRC) and Military Aircraft (MA) have been developing a method to predict the stress/strain response of aircraft structures in these extreme loading situations. The primary .consideration has been the requirement to create an acoustic fatigue design tool for dealing with combined static and dynamic loads, including thermally generated “quasi-static” loads. 1007 The resonant response of thin aircraft structures to aeroacoustic loading is generally in a firequency range which implies that, if defects form, they will quickly grow. Hence to be conservative, it is generally assumed that a component has reached its life when it is possible to find quite small defects by non-destructive evaluation techniques. Several different materials and construction methods are used in modern aircraft and so there are a number of possible failure criteria. In the case of metals, it is the presence of cracks larger than a certain size. For composites it can be the occurrence of either cracking or delamination. Degradation due to the presence of microcracks may be monitored by measuring the level of stiffness reduction which has taken place. This philosophy simplifies the type of stress analysis needed, because it is not necessary to model structures with defects present. Materials can be assumed to have simple elastic properties which remain unchanged throughout their lives. In consequence, it is necessary to know the fatigue behaviour in terms of a direct relationship between number of cycles to failure and the magnitude of the “nominal” cyclic stress, or strain, at a reference location. If considered important and capable of satisfactory treatment, the relationship can be modified to take into account material property changes due to the development of very small defects at points of stress concentration. For example, metal plasticity in the region of a small crack, could be included in an analysis of the stress distribution around a fastener hole. It is well known that plasticity reduces the peaks of stress which are predicted at defects by analysis which assumes perfectly elastic material behaviour. The technique developed at BAe for modelling high acoustic loads combined with possible thermal buckling uses the time domain Monte Carlo (TDMC) technique together with finite element analysis by proprietary FE codes. Response characteristics are predicted directly in the time domain using simulated random acoustic loadings. These may then be used in fatigue life estimations which employ cycle counting methods such as Rainflow counting. It is now practical to predict the vibrational response of stiffened aircraft panels without the necessity to assume a linear response, and without simplifying the spatial and temporal representation of the noise excitation. Since the technique uses proprietary finite element codes, quite large and complex models of aircraft structure can be analysed in a single run. Standard pre- and post-processor techniques are available to speed up generation of the finite element mesh and to display the stress/strain results. The initial development work was carried out by modelling the random vibration of a flat plate. For fully in-phase random loading at low noise levels the predicted response is predominantly single mode and at fhe frequency calculated by linear theory. However, as the decibel level is increased, the frequency of the fundamental rises due to geometrically non-linear stiffening. At veiy high dB levels the predicted response becomes multi-modal; the resonance peaks move to higher and higher frequencies and broaden. The effect of static loading on the response has been studied as part of these investigations to assist in the validation of the methodology being developed. Thermal Effects In some flight conditions it is possible for a panel to be buckled due to constrained thermal expansion and also be subjected to very noise levels at the same time. An example is when a ASTOVL aircraft hovers close to the ground for an extended period, panels which are initially curved, or thermally buckled panels may possibly be snapped through from one side to the other by a large increase in dynamic pressure. “Snap-through” can be potentially damaging to the structure of an aircraft if it occurs persistently, because the process is associated with a large change in the cyclic bending stress present at the edge of a stiffened panel. High performance aircraft must therefore be designed so that snap-through never occurs in practice. 1008 The dynamic response of curved panels or buckled flat plates is difficult to predict theoretically because of non-linear effects. The established acoustic fatigue design techniques, which are based on linear vibration theory, are only able to provide approximate predictions of the loading regimes in which particular panels might be expected to undergo snap-through. The TDMC method can be used to model non-linear multi-modal vibrations of stiffened aircraft panels which are also subject to quasi-static stress. In particular calculations may be carried out in the post- buckling regime. With this technique simulated random dynamic pressure loading, with measured or otherwise known spectral characteristics is applied to a curved, or post-buckled panel and the time domain response calculated. The magnitude of the dynamic loading may then be increased until persistent snap-through is observed in the predicted response. This gives the designer the ability to design out the potential problem by systematically altering the most important parameters in order to identify the critical regime. Fatigue Life Estimation Although acoustic fatigue is a complex phenomenon, it has been established that the life of a component mainly depends on its stress/strain history. The most important factors in this regard are the magnitude and frequency of the cyclic strain and the mean level of stress at the likely failure points. On this basis fatigue life can be estimated by carrying out the three stage operation illustrated in Figure 1. Stage 1: Determine Loads A determination of static design loads is relatively straightforward compared to a calculation of th.e full temporal and spatial dependence of the aeroacoustic pressure on a military jet in flight. This is an enormous task in computational fluid dynamics (CFD). Designers have to rely on experimental data which can come from measurements on existing aircraft or from scale model tests of jets, for example. Existing databases can be extrapolated if the circumstances are similar. Experimental noise data is usually in the form of power spectral density curves as opposed to time series fluctuating pressures, but either can be used, depending on the circumstances. Stage 2: Calculate Stresses The technique chosen to obtain the stresses clearly depends on how much knowledge there is about the expected loads. In the early stages of design analytic techniques would be used to establish approximate sizes and stress levels. However, later on when the design is nearly completion, finite element (FE) stress analysis can be used to model the effect of random acoustic loading on the parts of the skin which are likely to be severely affected. These calculations would, of course, be done including the effect of attached substructure. The established method of designing against acoustic fatigue uses a frequency domain technique which relies on the validity of linear vibration theory. The method forms the basis of a number of methodology documents published by the Engineering Sciences Data Unit (ESDU). Whenever there^^e large out-of- plane deflections the frequency domain method cannot be used because of the “geometric non¬ linearity”. From a strictly theoretical point of view such analyses have to be carried out in the time domain, although approximate methods are applied with some success. The established technique produces inaccurate results for curved panels, buckled panels and for panels under high amplitude vibration. Geometric non-linearity usually stiffens a structure in bending so there is a tendency to overestimate the stress levels using the frequency domain technique. This conservatism is clearly useful from the point of view of safety, but it can lead to possible “over-design”. Unfortunately this is not always the case when there are compressive static stresses present. The established method also fails if the phase of the noise varies significantly over the surface of the structure, which is the case in a number of aeroacoustic problems. The techniques under development are designed to overcome these problems. 1009 Stage 3: Estimate Fatigue Life In cases of random acoustic loading it is customary to assume that damage accumulates according to the linear Miner's rule. Fatigue life is determined from experimental data in the form of stress (or strain amplitude), S, versus number of cycles to failure, N. If a number of cycles, n, of stress/strain, S, occur at a level of stress/strain where N(S) cycles would cause failure then the fractional damage done by the n(S) cycles is n(S)/N(S). Various methods have been developed for obtaining n(S) from the stress (strain) response. If the excitation is stationary, ergodic and the response is narrow band random then the function n(S) can be shown to be in the form of a Rayleigh distribution and the damage sum can be evaluated from plots of root mean square stress (or strain) against number of cycles to failure. If the statistics of the response are not Gaussian then it is necessary to count the numbers of stress cycles from the time domain response and use constant amplitude S/N curves. It is now widely accepted that the best way of counting the cycles is to use the Rainflow method, [1]. The Loading Regime The loads on an aircraft may be conveniently divided into static and dynamic. Loads which vary only slowly are: a) Steady Aerodynamic Pressure Loading, b) In-Plane Loads transferred from “external structure”, and c) Thermal Loads due to Constrained Expansions. The rapidly varying loads are, of course, the aeroacoustic pressure fluctuations which originate from any form of unstable gas or air flow. This division is central to the methodology which has been developed because it enables the modelling to be carried out in two distinct phases. The, so called, static loads do vary, of course, but the idea is to separate effects which occur on a time scale of seconds from the more rapidly varying acoustic phenomena. The aim is to split the loads so that the quasi-static effects can be calculated in an initial static analysis which does not depend on a particular dynamic loading regime. Any aerodynamic pressure may be divided into a steady part and a fluctuating part. The natural place to make the cut-off is at IHz which means that epoch times for TDMC simulations are then of the order of a second. The epoch time must not be too short because of statistical errors, and it cannot be too long because this would invalidate the assumption of constant quasi-static loads. In practice, there is another constraint on the epoch time. The number of finite elements in the model coupled with the premium on cpu time places an obvious limit on the epoch time. Comparison of the Time and Frequency Domain Methods A flow chart comparing the two methods is given as Figure 2. The main difference between the two techniques lies in the representation of the dynamic loads. The FD method uses rms loadings and spectral characteristics, whereas the TD method uses the full time series loadings. Gaussian statistics are, de facto, assumed by the FD method, but this is not necessarily the case with the TD technique. Application of the frequency domain method requires that the response is dominated either by a single mode or a small number of modes. To determine whether or not this is the case in practice, a normal modes analysis must be followed by a determination of the amount of coupling between the excitation and each mode. This can be determined quite accurately even if there is a certain amount of potential non-linearity by computing the joint acceptances for each mode, which are overlap integrals of the mode shape functions with the spatial characteristics of the excitation. Normally these quantities will be dominated by a few of the low order modes. If there is significant coupling into more than one mode then it will be necessary to use the TD method instead. With the time domain technique it is possible to represent the dynamic loads in a way which models the convection of the noise field across the structure. Very complicated loadings can be applied to large 1010 models but in consequence it can be difficult to validate the results obtained, because they cannot be checked against anything other than test data which is itself subject to confidence levels. In addition it must be remembered that the TDMC results themselves are subject to statistical variability. Finally it should be noted that TDMC data must be used in conjunction with constant amplitude endurance data. Rms fatigue data can only be used with frequency domain results. Time Domain Finite Element Modelling Until recently, the majority of finite element analyses were applied to static loading conditions or “low frequency normal modes analysis”. The method involves the use of an implicit code to invert in one operation, a single stiffness matrix, which can be very large. The general availability of extremely fast super-computers has now made it possible to carry out large scale non-linear dynamic finite element modelling using explicit FE codes. These codes use very similar types of element formulation to the implicit ones, e.g. shells, solids and bars, but the solution is advanced in time using a central difference scheme. One potentially very useful capability of time domain modelling is the application of acoustic pressure loadings which vary both in the time and spatial domains. If the spectral characteristics are known, either from test or from other modelling it is possible to generate samples of random acoustic noise and apply these directly to the finite element model as a series of “load curves”. The technique for determining time series noise was developed by Rice [2] and Shinozuka [3]. They showed that homogeneous Gaussian random noise can be generated from the power spectral density as a sum of cosine functions with different frequencies and random phase. Noise can be temporally and spatially correlated noise by deriving phase differences from cross spectral functions if they are known. The TDMC method can be quite costly in terms of central processor unit (cpu) time because the solution must be recalculated at each point in time. To reduce execution times, the explicit codes employ reduced numbers of volume integration points in the finite element formulations. However in this work cpu times are extended because long epoch times are required to ensure adequate statistics. It can take more than 24hrs to obtain a solution over a half second epoch if there are a few thousand elements in the model. Hence there is always a practical limit to the size of a particular time domain finite element analysis, (TDFEA). If the loading and geometry are not too complicated, the frequency domain method of analysis can be tried initially to gain more understanding of the nature of the response in an approximate way. In some cases the vibrational response regime must be considered carefully to decide whether TDFEA is really necessary. These may be situations where the non-linear effects are only moderate. It would be ideal if the full dynamic response of an aircraft could be determined with a fine mesh model in one huge operation, but experience has shown that this requires too many elements. It is possible to construct frill models with reduced stiffness using superelements, enabling flutter and buffet to be studied, because these are essentially low frequency phenomena. However, in time domain analysis it has been found that models containing a large amount of detail, such as fasteners and individual composite material plys, require a great deal of cpu time. To progress we must devise some strategies to overcome this situation. Since a full TDFEA can only handle a part of the aircraft structure, it is very important that loads external to the area under consideration are properly taken into account. This is cmcial to the success of this type of modelling as it is to all finite element modelling. The most important parameter in any time series analysis is the time step. This is determined by the velocity of sound waves in the structural material, and is generally of the order of a/v where a is the shortest element dimension and v is the velocity of longitudinal sound. A small time step is therefore required when the elements are small and the velocity of sound is large. For an aluminium model with 10mm square elements the time step is about 1.6ixs. Hence a TDMC run with a half second epoch time needs about a half a million steps. A simple 5000 shell element calculation on a Cray C94 would take approximately 10 hours. 1011 Dynamic FE models of aircraft structure can be constructed in many ways, using shell elements, beam elements and/or solid elements. Special elements exist for damping and for sliding interfaces. Joints can be modelled with sliding interfaces, or with short beams, or just with tied nodes. Fasteners can be modelled with small solid elements, with short beams or with tied nodes, also. Unfortunately, however short beams and small solid elements cause a dramatic lowering of thC time step. For example, if the smallest fastener dimension is, say 3mm, the time step will have to be reduced to about O.Sfis if solid elements are used in the model. The effect on cpu time is such as to make the calculations impractical. Sliding interfaces are an efficient way to model skin/substructure contact in explicit analyses, but it is important to choose the algorithm carefully because some techniques can consume large amounts of cpu time. The best practical way of representing stiffened aircraft panels for TDMC analyses is considered to be with four noded shell elements simply tied together at their edges. A number of efficient shell formulations are avail-able and meshes can be rapidly produced from the design geometry. Of course, such models cannot be expected to produce highly accurate stress data in the region of small features but this aspect has to be sacrificed in the interests of achieving statistically significant amounts of time series data. To improve the accuracy of stress predictions in the neighbourhood of stiffeners etc., it will be necessary to couple TDMC analyses with fine mesh static analyses. The Generation of Time Series Data A number of factors must be borne in mind when generating time series data for TDMC calculations. It is important to consider carefully the frequency range and number of points which define the load spectrum in conjunction with the epoch time and number of points on the time series. The Nyquist Criterion [4] states that the time increment must be less than or equal to one over twice the upper frequency on the power spectral density curve. For the sake of argument, take the upper frequency to be IkHz. This means that the time increment must be less than 500ps. A more conservative time increment is based on the requirement to represent the dynamic response of the structure as accurately as possible over a full cycle. Assuming a resonant frequency of 500Hz, which is perhaps near the limit in practice, and 10 points per cycle which is more than sufficient, the lower limit on the time step works out at about 2C)0|is. Taking all these factors into consideration, the number of points on the spectrum curve should be of the order of 1000 and there should be between 1000 and 5000 on the time series. Longer epoch times can be used but for reasons of practicality and statistics it is better to run more than one short epoch simulation rather than one long simulation. Explicit FE modelling frequently requires that the time step be smaller than 200jj,s. In the example given above the time step required by the explicit code was 1.6fis. Under these circumstances the random noise could be defined with a smaller time increment, but going to this level of effort has been found to produce no measurable change to the calculated response. Static Initialisation There are two possible ways of dealing with the effect of static loads in TDMC modelling. Firstly the complete analysis can be carried out using the explicit code. To do this it is necessary to apply only the static loads to the model and run the code until equilibrium is reached. By introducing a high level of artificial damping the stresses created can be relaxed in a relatively short period of time. The time required depends on the lowest resonant frequency of the structure and the size of the smallest element in the model. This facility is termed “dynamic relaxation”. The alternative is to make use of another facility in the explicit code called “static initialisation”. The deformed shape and stress state of the structure with just the static loads applied are first obtained very quickly using an implicit code. The solution for the stressed state is then initialised into the explicit code prior to the application of the dynamic loads. Dynamic relaxation may be used to smooth out any differences between the models. 1012 Damping Representation Vibrating aircraft structures are damped by several mechanisms, for example friction at joints, re¬ radiation of acoustic waves, and energy loss in viscoelastic materials It is difficult to generalise about the relative importance of each damping process in practice. Also reliable quantitative data is not available in sufficient detail to justify the inclusion of complex models of damping into the TDMC analyses. Test results on vibrating stiffened aluminium panels tend to show that the damping is best approximated by a combination of mass and stiffness proportional coefficients. There is a range of frequencies in which the damping ratio can be considered to be roughly constant. Until more detailed experimental data are available the most expedient approach is to assume a nominal value for the global damping ratio which does not change with frequency. Over the years it has become standard practice to assume a damping ratio of about 2% for fastened aluminium structures. Equivalent Linearisation There are some loading regimes in which the non-linear response to high levels of random acoustic loading can be approximately found using a linearisation technique combined with a frequency domain analysis. The basic idea is to replace the non-linear stiffness term in the general vibration equation by a linear term such that the difference between the rms response of the two equations is minimised with respect to a shifted fundamental resonance frequency. If an approximate equation for the non-linear stiffness is known then it is possible to derive an expression for the shifted “non-linear” resonance frequency. The rms response to random acoustic loading may then be found by combining the Miles/Clarkson equation with some form of static geometrically non-linear analysis. References to this technique are Blevins [5], Mei [6] and Roberts & Spanos [7]. Where the geometry is complex the most appropriate form of analysis is clearly finite element analysis. Implementation and Validation Studies The stress analysis work described in this paper has been undertaken using MSC-NASTRAN and LLNL-DYNA. NASTRAN is a well known implicit finite element code which is capable of handling very large numbers of elements. It has been developed very much with aerospace structural analysis in mind. It is basically a linear analysis code, although there are a large number of adaptations to deal with non-linear problems. It can also function as a dynamic code, but is much slower than DYNA in this mode because it basically needs to solve the complete problem at each time step. The non-linear features which are most relevant to the type of stress analysis being discussed here are those concerned with geometric non-linearity. Geometric non-linearity is treated by dividing the load into a series of steps, obtaining the solution incrementally. In this work, the code has been used for linear and non-linear static analyses and for normal modes analyses. DYNA is an explicit finite element code originally developed for the calculation of the non-linear transient response of three dimensional structures. The code has shell, beam and solid element models and there are a large number of non-linear and/or anisotropic material models available. DYTMA was developed primarily for the modelling of impact and there is no limit, as far as the code is concerned, on the size of finite element model which can be analysed. Many of its advanced features relate to impact modelling and are not required for this work. One useful feature, however is the laminated composite material model based on the equivalent single layer approximation. This code has been used for the TDMC calculations presented in this paper. Finite element models for the stress analyses were produced using MSC-PATRAN, It has a wide range of geometry and mesh generation tools and now has built in interfaces for both NASTRAN and DYNA. The element definitions are compatible with both codes and it is a simple matter to toggle between the two codes by changing the analysis preference. Not all the features of DYNA are supported and some of the parameters must be set by editing the DYNA bulk data produced by PATRAN. NASTRAN results were post-processed using PATRAN. DYNA results were post processed using TAURUS, which is faster and easier to use than PATRAN for this task; Some special in-house codes have been written to generate random acoustic noise from power spectral densities, as described above, 1013 and to post process time series output from TAURUS. One of the codes incorporates a fast Fourier transform (FTT) routine to determine spectral responses from the DYNA time series predictions. These codes are covered under the generic title “NEW-DYNAMIC”. TDMC Calculations on a Simple Flat Plate Calculations have been performed on a very simple model to implement the TDMC technique and develop the in-house software referred to above. A PATRAN database was constructed representing a simple flat plate, 350mm x 280mm x 1.2mm thick as an array of shell elements 34x28. For simplicity the boundary conditions were taken either as simply supported or clamped. There are a number of alternative shell element formulations available in DYNA, [8]. The Hughes-Liu shell was used initially because of its good reputation for accuracy, but later a switch was made to a similar, but slightly faster shell element, called the YASE. It was found that equally satisfactory results could be obtained more quickly using this element. Analyses without Static Loads A series of DYNA calculations were carried out with a fiiUy correlated random acoustic pressure load with a flat noise spectrum between OHz and 1024Hz. Investigations were carried out into the effect of varying the sound pressure level, the epoch time, the mesh resolution, the damping coefficient and the stochastic function. Figure 3 shows the displacement response of the central node of the model for a sound pressure level of 115dB (about 12Pa rms), simply supported edges and mass proportional damping set so that the damping ratio was equal to 2% at the fundamental (1,1) resonance of the plate. The corresponding spectral response is shown in Figure 4. shows a sharp resonance peak at a frequency of 61.0Hz which is very close to the theoretical frequency of the (1,1) mode for the simply supported plate. The in-phase loading means that only the modes with odd numbered indices are excited. I^e peaks corresponding to the (3,1) and (1,3) modes are, however, not visible on the plot because they are too small. It may be concluded from these results that the behaviour of the plate at these pressure level is well within the linear regime. An investigation into the behaviour of the rms displacement response as a fijnction of SPL was carried out by increasing the loading incrementally from 75dB (0.12Pa rms) to 175dB (12kPa rms). The results are shown in Figure 5. Also shown are theoretical predictions obtained using the Miles/Clarkson formula with NASTRAN linear and non-linear analyses as explained above, see below for discussion. The statistical variation of the results was investigated by repeating a half second epoch TDMC run ten times with different samples of flat spectrum noise. It was found that the standard error of the rms response was about 16%. A second set of ten repeats were carried out with the epoch increased to 2.5s. In this case the standard error reduced to roughly 8%. From the theory of stochastic processes, it can be shown that the standard error is inversely proportional to the square root of the epoch time. On this basis therefore the ratio between the standard errors should be equal to the square root of five, or 2.23. From the analyses this ratio is about 2. Further runs established that these results are not affected by the vibration amplitude, even when the response becomes non-linear. Cautiously therefore, it can be concluded that the variance of the TDMC results is unaffected by non¬ linearity of the response. This is an important finding because it builds confidence in the technique. In many practical situations it may be necessary to rely on just one simulation and an appropriate factor of safety. It can be quite time consuming to carry out a large number of repeat TDMC simulations. The level of variance would be first established by repeating one load case a number of times, before confidently applying it to the results of other load cases. Comparisons with Linearised Theory The linear theory of plate bending, [9], leads to relationships between the central deflection, w, of a rectangular plate and a uniform static pressure load, Pstat which take the following form. Psutab = kcffW (1) 1014 where a and b are the length and breadth of the plate, and k^ff is an effective stiffness parameter which is a function of the modulus of rigidity of the plate and the edge boundary conditions. For the plate studied k^ff is about 30N/mm for the case of simply supported edges and lOON/mm for clamped edges. The above equation only holds, however, at very low amplitudes, as can be seen from Figure 6. This compares geometrically non-linear NASTRAN predictions with the linear ^eory. Curves are shown for both simply supported and clamped boundary conditions. The finite element results show the characteristic hardening spring type of non-linearity. At higher amplitudes the dynamic behaviour may be approximately predicted using “equivalent linearisation” theory, which assumes that the response remains predominantly single mode, but with a resonant frequency which rises as the stiffness of the structure increases. When the deflection is large the static force-deflection relationship can be written as the sum of a linear stiffness term and a cubic non-linear term: psiatab = kw(l+pw^) (2) where a b is the force, k is the linear stiffness. The equation is written with the leading term factored out to emphasis the point that p. is a constant which is small compared to the rms deflection. In the limit of small w we can expect the pw^ term in the brackets to be negligible compared to one, which means that the k in this equation must be the same as kcff above. Equation 2 was fitted to the NASTRAN results shown in Figure 6 to find the best fit values of k and p. Table 1 shows the results compared with the effective stiffness calculated from linear plate bending theory. It can be seen that the theoretical stiffness is almost identical to the best fit k from the non-linear finite element analysis. With reference to the results in Figure 5, it is obvious that the nature of the response is strongly dependent on the amplitude of the vibrations. For rms displacements up to about 4% of the plate thickness the behaviour was completely linear. For displacements between 4% and 150% of plate thickness, the response was essentially single mode dominated but the level could not be predicted by the Miles/Clarkson approach. The “equivalently linear” solution does, however, agree with the DYNA result up to a displacement of about l.8mm. The linearisation approach cannot be expected to be correct for displacements above about 1.5 times plate thickness. Above this point the response predicted by DYNA was multi-modal and strongly non-linear. The equivalently linear predictions departed considerably from the DYNA results when the vibration amplitude was very high. It was also observed that the frequency response peaks became increasingly noisy for higher pressures, representing the increased level of non-linearity in the plate vibrations. The increase in the frequency of the fundamental mode with acoustic pressure, as calculated by DYNA, is shown in Figure 7. Predictions from equivalent linearisation theory and from the theory of Duffing's equation are also included, see Nayfeh & Mook [10]. t„, = f(l + 3nw™,Y’ Equiv. Lin. Pred. (3) f„, = f(l+(3/8)w„,^) Duffing's Eq. Pred. (4) The DYNA results lie mostly between the two theoretical curves, agreeing particularly well with the results of equivalent linearisation theory up to around 700Pa (151dB rms). The level of agreernent obtained shows that the frequency response behaviour of the DYNA model is similar the theoretical predictions, providing an independent check on the results. As might be expected, at around 700Pa the agreement begins to breakdown, since the linearisation theories are not valid for deflections which are significantly greater than the plate thickness. It may be concluded, however, that the effect of geometric non-linearity at high amplitudes is being computed by DYNA in a reasonably accurate manner. A detailed comparison with experimental data is needed to determine the accuracy of the DYNA response predictions themselves. 1015 Due to the increasingly irregular shape of the frequency response functions derived from the DYNA time series predictions at high acoustic loads, it was not possible to calculate very accurate peak widths for pressures above approximately 135dB (I20Pa rmsX Figure 8 shows that the width of the peak increased with increasing acoustic pressure, but not in a regular manner. When the damping is mass proportional, equivalently linear theory predicts that the width of the peak should remain unchanged as the pressure rises. This is because the geometric stiffening effect of rising acoustic pressure exactly cancels the effect of a smaller damping ratio at the higher resonant frequency. This graph shows this as a horizontal straight line at 2.44Hz. The DYNA result is closer to the type of behaviour observed experimentally where the width of the peak generally increases with increasing the sound pressure level. Analyses with Combined Loads Further work was conducted with static loads superimposed on different levels of random acoustic loading. These calculations were done using the coupled NASTRAN-DYNA approach outlined earlier. That is to say the deformed geometry was obtained by applying the static loads to a NASTRAN model, with the results being initialised into DYNA and dynamically relaxed before the dynamic loading was applied. Calculations were performed with compressive in-plane loading, static pressure loading and thermal loading. With the exception of the thermal runs, the boundary conditions used in these runs were identical to clamped, except that symmetrical in-plane movement of the edges was permitted. We have called these conditions “semi-clamped”. It has been found that the fundamental resonant frequency of the plate without static loading is only reduced by a very small amount if the appropriate in-plane degrees of freedom are released, see Figure 9. These boundary conditions are actually closer to those which exist in reality when a panel in built into a larger structure. Figures 10-14 show results of some of the analyses which have been carried out. They give time series data along with spectra responses calculated by the in-house post-processing code. Numerical data derived from these results are summarised in Tables 3-5. A series of analyses have been carried out with compressive in-plane loads equal to one third of the theoretical buckling loads in compression. For the plate used, the forces per unit side length were - 3.46N/mm in the x-direction and -5.46N/mm in the y-direction. The results of one analysis are shown in Figure 10. It has been found that the response remains dominated by the fundamental (1,1) mode as long as the plate is unbuckled and the SPL is low. The softening effect of the compressive loads on the frequency agreed quite well with Rayleigh-Ritz predictions, [5], up to an SPL of ?dB. At higher sound pressure levels, the DYNA results reflected stiffness changes which were greater than those predicted by the theory. The- same was found in the case of tensile loading. It is believed that these differences are due to approximations built into both the Rayleigh-Ritz theory and the DYNA code. Figure 11 shows the results of a calculation with a superimposed normal pressure. The magnitude of the pressure, 700Pa, was chosen so as to provide an example of “post-buckled” analysis. This size of pressure causes the plate to bow out in the centre by about 0.6mm. It is well known that in the post- buckling regime the random response of a plate depends upon the magnitudes of both the static and dynamic loads. In this case the static loading was large compared to the applied dynamic loads and “snap-through” did not occur. The plate simply vibrated about its statically deflected position in the fundamental mode with a slightly increased frequency. To provide a test of the DYNA thermal stressing capability, and to carry out an investigation into “snap-through”, several analyses were carried out with a uniform temperature rise of lOdeg C applied to the plate with clamped edges. This is quite sufficient to cause buckling because the resulting compressive biaxial stress, c, is well above the buckling level, Gb- If f is the frequency of the fundamental and J is a constant equal to 1.248 because of the clamped boundary condition, the two stresses can be determined approximately from a = EaT/(l-v^) (5) Cb = 4pa^f2/J (6) 1016 where E, a, v and p are Young's modulus, coefficient of thermal expansion, Poisson's ratio, and density respectively. Using these formulae we find c = 24MPa and Ob = 14MPa. Analyses were carried out with several different levels of dynamic load. The results of three of the calculations are shown in Figures 12,13 and 14. It was found that the threshold for snap-through occurred at an acoustic load of about IkPa, see Table 4. Below this level the mean deflection, w, is a function of the static load alone, equal to about 2.8nun (the negative sign indicates that the plate has bowed in direction of negative z). At higher SPLs the mean deflection reduces because the plate snaps backwards and forwards between positive and negative z. The calculated response spectra for these higher level runs, show an additional peak at a very low frequency, ie less than lOHz. This is an artifact caused by the snap-through since the fundamental resonance of the clamped plate is at 113Hz. Figures 10-14 all show probability density functions derived from the time series data. The fluctuations on these plots are caused by the smallness of the epoch time. In all cases, except for the thermal calculations with the two largest acoustic loads, it can be seen that the functions are basically Gaussian in shape. It may therefore be concluded that it is reasonable to assume that the response of a plate in the post-buckled region is Gaussian unless there is a large amount of snap-through. Discussion The work described is the starting point for investigations and validations using more complex FE models. Further work has been carried out using models including curvature, sub-structure and detailed features. It is difficult to validate the predictions obtained from such models by comparing with test data because the results themselves are open to interpretation. It has been found that the predicted stress levels are closer to the test results when the chosen location is away from any small features. The lack of good agreement in the neighbourhood of the features can be explained by the relatively coarse mesh used in the dynamic models. The overall level agreement was much better than that between predictions based on linear or equivalently linear theory and test. On the basis of experience, the latter tend to over¬ predict by upwards of factors of two and three. From this work it has been found that the DYNA predictions tend to be greater than test by amounts which vary but are generally much less. The average over-prediction was about 40% with a significant change as a function of location. TDMC runs can take a significant amount of computer time to carry out and it is believed that to make further improvements the technique should be combined with detailed stressing using static finite element analysis. Inaccurate results can be obtained if the boundaries of the part of the structure under analysis are not properly restrained. In the case of models of aircraft panels this may significantly affect the resonant frequencies which in turn affects the level of calculated dynamic stress. In-plane loads on a panel, perhaps due to thermal stressing, can alter the fundamental by as much as 100-200Hz. Looked at from a theoretical point of view, the only way to solve this problem is to construct a second, coarse model of the component, along with some of its surrounding structure. An initial calculation can then be carried out with this model in order to obtain the loads and boundary conditions for subsequent application to the original model. The dynamic phenomenon of “snap-through” cannot be modelled using existing methods and so the TDMC / finite element technique offers the engineer a way to determine where the likely regions of unstable vibration are located in circumstances where the structure is complicated by attachments etc. Conclusions This paper has sought to explain how time domain finite element modelling can be used to assist in the design of aircraft against acoustic fatigue. Although the technique is computationally intensive, it does have a place in the effort to understand complex vibrations, such as the response of structures to spatially correlated jet noise excitations, or interactions between high sound pressure levels and thermal loads. The work at BAe is continuing in an attempt to provide the analyst with a greater ability to determine dynamic stress levels in advanced structures with complex loadings. 1017 References 1. Dowling N. E. , Fatigue Prediction for Complicated Stress Strain Histories, J Materials 1, 71 (1972). 2. Rice, In Selected Papers on Noise and Stochastic Processes, Ed N Wax pplSO, Dover New York (1954). 3. Shinozuka M. , Computers and Structures, 2, 855, (1972). 4. Bendat J. S. and Piersol A.G. , Engineering Applications of Correlation and Spectral Analysis Wiley (1990). 5. Blevins R.D, , An approximate method for sonic fatigue analysis of plates and shells, J Sound and Vibration, 129, 1, 51 (1989). 6. Mei C. and Paul D.B. , Non Linear multi-modal response of a clamped rectangular plates to Acoustic Loading, AIAA Journal, 24, 634, (1986). 7. Roberts J. B. & Spanos P.D , Random Vibration and Statistical Linearisation, Wiley, (1990). 8. Whirley R.G. and Engelmann B.E. , DYNA3D: A Nonlinear, Explicit, Three Dimensional Finite Element Code for Solid and Structural Mechanics — User Manual, Lawrence Livermore National Laboratory, UCRL-MA-107254 Rev. 1, (Nov 1993). 9. Szilard R., Theory and Analysis of Plates, Prentice Hall, New Jersey. 10. Nayfeh A.H. & Mook D.T. , Non Linear Oscillations, Wiley (1979). 1018 Parameter lUQgnnggll HISSH Linear Theory NASTRAN NASTRAN Simply Supported 30.0 30.7 1.09 Clamped 104. 101. 0.266 Table 1: Values of Parameters fitted to NASTRAN results compared with the linear theory values. Static Loads Acoustic Load Rrms (Pa/dB) Theory ] 1 DYNA (N.mm N, (N.mm Freq (Hu) Wfim (mm) ■SI Wrms (mm) None 1.2 (95.6) jnggi 0.00290 114 0.01.2 -3.46 1.2 (95.6) 0.00316 94.1 0.00313 -3.46 643.5 (150.1) 1.69 115 1.08 -3.46 1.2 (95.6) 68.33 0.00359 68.1 0.00368 -3.46 700 (150.9) 68.33 2.09 103 1.42 Table 2: Summary of results of calculations with random acoustic loading superimposed on compressive in-plane loads. Static Load p(Pa) Acoustic Load (Pa/dB) DYNA None 1.2 (95.6) 114 0.0000 0.01.2 700 None — -0.646 — 700 12 (115.6) 115 -0.655 0.0189 700 700 (150.9) 134 -0.477 1.28 5 k None — -2.89 — 5 k 12 (115.6) 179 -2.93 0.0152 5 k 700 (150.9) 173 -2.70 0.999 Table 3: Summary of results of calculations with random acoustic loading superimposed on static pressure loads. Temperature T (<»C) Acoustic Load (Pa / dB) 1 DYNA 1 None 1.2 (95.6) 114 0.00256 10 None — -2.85 — 10 1.2 (95.6) 234 -2.86 0.0140 10 700 (150.9) 219 -2.80 0.197 10 Ik (154.0) 9.01 2.47 1.02 10 1.2k (155.6) 5.01 -0.717 2.24 10 1.5k (157.5) 3.00 -0,376 2.20 10 2k (160,0) 92.2 0.0185 1.95 10 4k (166.0) 195 0.0568 2.08 Table 4: Summary of results of calculations with random acoustic loading superimposed on a thermal load. 1019 Figure 2: Flowchart Illustrating the Frequency and Time Domain Techniques (uiUi),ueujeoB,ds,a ^ O Displacement of central node for 12Pa rms acoustic pressure, 1 . DYNA Figure 4: Spectral Response of the fiat plate corresponding to Fig 3. Central Node for 12Pa rms pressure, DYNA calculation. 3 2.5 2 1 o ^ 1 0.5 0 Figure 5: Rms Central Deflection of the plate versus sound pressure level Comparison between DYNA results and linear theory. 3 2.5 2 ■£ £ E I I o 1 0.5 0 Figure 6: Central deflection of the plate versus pressure, NASTRAN calcula¬ tions compared to linear theory. 500 1000 1500 2000 2500 3000 3500 Static Pressure (Pa) rms Pressure (Pa) 1022 Figure 7: Variation of fundamental frequency of the (1,1) mode with rms pressure, Comparison between DYNA and theory. Damping 2%; Stoctiastic Fn #1; 3Sx29 Nodes 0 - 1 - 1 - 1 - 1 -- 0 200 400 600 800 1000 1200 rms Pressure (Pa) 130 140 145 150 153 155 SPL (dB) Figure 8: Variation of width of the (1,1) mode resonance peak with rms pres¬ sure, Comparison between DYNA and theory. 1023 Figure 9: DYNA model predictions for random vibration of the plate with semi-clamped boundary conditions. Figure 10: Random Vibration results with compressive load in the y-direction of -3.46N/mm and SPL of l.OPa. Figure 11: Random Acoustic Loading of SPL=12Pa superimposed on a static pressure of TOOPa. 1024 Figure 12: Random acoustic loading of SPL=700Pa superimposed on a thermal load of 10 deg, clamped edges. — No Snap Through Figure 13: Random acoustic loading of SPL—2kPa superimposed on a thermal load of 10 deg, clamped edges. —Nearly continuous snap-through. Figure 14: Random acoustic loading of SPL=4kPa superimposed on a thermal load of 10 deg, clamped edges. —Dominant acoustic load. 1026 SYSTEM IDENTIFICATION II ROBUST SUBSYSTEM ESTIMATION USING ARMA-MODELLING IN THE FREQUENCY DOMAIN by U. Prells, A. W. Lees, M. 1. Friswell and M. G. Smart, Department of Mechanical Engineering of the University of Wales Swansea, Singleton Park, Swansea SA2 8PP, United Kingdom ABSTRACT This paper reflects early results of the research on modelling the influence of the foundation on the dynamics of the rotor. The foundation is connected to the rotor via journal bearings. Dynamic models exist for the subsystems of the rotor and of the bearings; the first is reliable but the latter is uncertain. The foundation model is unknown and has to be estimated using rundown data.These are measured responses of the foundation at the bearings due to unbalance forces of the rotor which are assumed to be known. Uncer¬ tainties in the bearing model will be transfered to the estimated foundation parameters. The main scope of this paper is to introduce a method which enables the decoupling of the problem of model estimation and the problem of the influence of the bearing model uncertainty. The influence of changes in the model of the bearings on the estimation of the foundation model is mainly due to the sensitivity of the computed forces applied to the foundation at the bearings. These are used together with the associated measured responses to estimate the foundation model in the frequency domain. Using an ARMA model in the frequency domain it is possible to estimate a filtered foundation model rather than the foundation model itself. The filter is defined in such a way that the resulting force has minimum sensitivity with respect to deviations in the model of the bearings. This leads to a robust estimation of the filtered model of the foundation. Since the filter can be defined in terms of the models of the rotor and of the bearings only, the problems of estimating the foundation’s influence and of the sensitivity of the estimates with respect to the model of the bearings are decoupled. The method is demonstrated by a simple example of a single-shaft rotor. Even if the errors in the bearing model are about 50 % the relative input and output errors of the filtered foundation model are of the same magnitude as the round-off and truncating errors. 1 INTRODUCTION An important part of a machine monitoring system for fault diagnostics of 1027 a turbo generator is a reliable mathematical model. This model includes the subsystems of the rotor, the bearings and the foundation. The model of the rotor represents the most reliable knowledge, the model of the journal bearings is uncertain, and despite of intensive research it is not yet possible to define a model for the foundation which refiects the dynamical contribu¬ tion to the rotor with sufficient accuracy. The first step to determine the contribution of the foundation on the rotor’s dynamic performance is the estimation of a reliable foundation model. Rundown data are available, i.e. displacements ufb{<->^) ^ of the foundation at the bearings which are due to an unbalance force /c/(a;) G of the m-shaft rotor, given at discrete frequencies a; € := {wi, ■ • •, wm}? and this data may be used to estimate the foundation model. A com¬ mon method ([1],[2],[3],[4],[5]) is to estimate the unknown dynamic stiffness matrix 6 i^4mx4m foundation at the bearings using the in¬ put/output equation F{uj)upb{(^) = (1) where the force fpB of the foundation at the bearings can be expressed by dynamic condensation in terms of the data ups, fu and in terms of the dynamic stiffness matrices Ar,B of the rotor and the bearings respectively yielding fpB = -Bufb + [ 0 , B]A~^q ^ ^ . ( 2 ) Here the dynamic stiffness matrix Arb of the rotor mounted on the bearings is partitioned with respect to the n inner degrees-of-freedom (dof) of the rotor and to the 4m connecting (interface) dof ^RB Aru Arib Arbi Arbb + B (3) The non-zero components of the force fpi G (D” in eq. (2) of the inner part of the rotor are the components of the unbalance force /y, i.e. introducing the control matrix Su € ^ dynamic stiffness matrix Ar of the rotor is given in terms of the matrices of inertia and stiffness which are defined by modal analysis and by its physical data given by the manufacturer. Each of the m shafts of the rotor is connected to the foundation usually via 2 journal bearings. Since the dynamic stiffness matrix B of the journal bearings represents a model for the oil film it consists of connecting dof only. It can be shown that B is block diagonal B = 0 0 (4) containing the dynamic stiffness matrices Bi = Ki+ jujDi.i = 1, • • • ,m, of the m bearings. Ki, Di are the matrices of stiffness and damping respectively 1028 which result from linearisation and are in general non-symmetric and non¬ singular. Eq. (1) is then used to estimate the foundation transfer function F{u). This has been discussed in several papers ([1],[2],[3],[4],[5]). Lees et al. [3] pointed out that fpB is sensitive with respect to deviations in the model of the bearings over part of the frequency range. This sensitivity is transfered to the model estimates. In this paper a method is introduced which enables the decoupling of the two problems of model estimation and of sensitivity of the foundation model with respect to the model of the bearings. The basic idea of this method is to estimate a transfer function H{u) which maps the displacements ufb{^) to a force /^(w) rather than the force i-e. H(u)ufb{^) = ( 5 ) In extension of the earlier method the force //f(a;) can be chosen to be of minimum sensitivity with respect to the model of the bearings. This robust estimated transfer function H{uj) is related to that of the foundation F(uj) by a transformation P{u) H{lo) = PMF(a;), (6) which of course retains the sensitivity with respect to the model of the bearings. But since P{uj) only depends on the models of the rotor and the bearings in the case of a modification within the model of the bearings no new model estimation has to be performed because this has been done ro¬ bustly with respect to such model changes. 2 THE OPTIMUM CHOICE OF THE FORCE VECTOR As stated in Lees et al. [3] the sensitivity of the force /fb with respect to B is mainly due to the inversion of the matrix Arb in eq. (2). It can be shown that the condensation method of estimating the force /fb results from the special case of eliminating the last 4m rows of the matrix W := Arii ^RBI 0 Arib ArbB -b P ^ ^(n+8m.)x(n+4m) ( 7 ) which can be written as Arb = T'^W ^ (l^(n+4m.)x(n+4m) defining the selecting matrix of the master dof as T := [ei, • • • , € R(-+8n)x(n+4m)^ (S) 1029 where in general en denotes a unit vector of appropriate dimension contain¬ ing zeros everywhere but in the nth place. In extension to the force fpB defined in eq. (2) for an arbitrary selecting matrix T € the condensation leads to a force fn given by j-^LT In+Sm-W{T'^Wy'T^ ” *■ V *' .. . =:P ( Bufb V } ( ^ \ 0 V -^4m J fpB- ( 9 ) Here T-^ € denotes the matrix which selects the slave dof, and in general In denotes the unit matrix of dimension n. Indeed, inserting the special choice of T from eq. (8) into eq. (9) leads to the sensitive force Ih = fpB as defined in eq, (2). The reason for the sensitivity of /fb is that the subsystem of the rotor has low damping. Near the resonance frequencies of Arb its large condition number depends sensitively on B. Thus the sensitivity of Jfb with respect to B is due to a large condition number of Arb • Let T denote the set of all possible selecting matrices, i.e. r := {[ei..--',ew4„]:e4elR"+*“. l<4<n + 8m, V /c = 1, • •«,n +4m}, (10) One criterion for an optimum choice of the force Jr may be formulated as the following minimisation problem: Criterion 1: The optimum choice is the solution of mmcond{T^W), (11) where W is defined in eg. (7). A low condition number is necessary but not sufficient in order to provide a low sensitivity of the force /h* Therefore a numerical test can be applied using stochastic deviations in the bearing model. Let A A consist of uniform distributed non-correlated random numbers with zero mean values and variances equal to 1/3 for alH G 1, ■ • *, 2m. Define ABi = ABfisi, Ti) := siAKi -h A A, (12) where the positive scalars si.Ti control the magnitude of the random error of the i-th bearing model. Thus, the error AB = AB{s,r) of the bear¬ ing model is well defined for s := (si, • ■ •, 52771 )"^ and r := (ri, • • • ,r 2 mV■ 1030 Regard the force fn = f{uj,T,AB) as a function of the selecting ma¬ trix T and the bearing model error AB. For I random samples AB{k) = AB(r{k),s{k)),k == calculate for each frequency a; G the up¬ per and lower bounds for the real and imaginary part of each component /i, ^ = 1, • • •, 4m of the force vector /, i.e.: //Lax(“.U ;= max Re {/i(a;,T, AB(/c))} , (13) := min Re {fi{u,T, AB{k))} , (14) := ^max^Im {/i(a;,T, AB(/c))} , (15) fLni^.T) := min Im AB{k))} . k=l,"-,l (16) Defining the force vectors /max(w,T) ■■= T)+j- T), (17) /mm(w,T) (18) the second criterion can be formulated as a minimax problem: Criterion 2: The optimum selection is obtained from 4m minmax^l/imax(w,r) - (19) TST uj£u . - 2 = 1 Before the method outlined is demonstrated by an example some aspects of the mathematical model of the foundation and methods for its estimation based on the input/output equation (5) will now be considered. 3 ESTIMATION OF THE FOUNDATION MODEL The purpose of this section is to estimate the unknown foundation model represented by the matrix F{uj) = Afbb{^) - Afbi{^)Af]j{uj)Afib{^)‘ (20) This expression results from dynamic condensation of the dynamic stiffness matrix of the foundation Af Afbb Afib Afib Afh ( 21 ) 1031 which is partitioned with respect to its inner dof (index I) and those dof coupled to the bearings (index B). For viscous damped linear elastomechan- ical models the dynamic stiffness matrix Aj?(a;) of the foundation is given by ^ Af(uj) := ( 22 ) i=0 The matrices Ai are real valued and represent the contributions of stijffness, damping and inertia for z = 0,1,2 respectively. In this case the identifica¬ tion of the foundation model requires the estimation of the three matrices Ai which are parameterised by introducing dimensionless adjustment pa¬ rameters aik € IR, for all /c = 1, * • •, ^ = 0,1,2 (see for instance [6] or [7] ). Those parameters are related to given real-valued matrices Sik by Ni Ai{ai) Jk=i Writing the adjustment parameters as one vector a"’’ ;= (af,aj’,aj) G IR^, p := ATq + Ni + W 2 , the estimation of the foundation model is equivalent to the estimation of the parameter vector a. The dynamic stiffness matrix of the foundation becomes a nonlinear function of this parameter vector ^"( 0 ;) =F[(jj^a) — Afbb[^-)0) “ Afb7(ci;j a)Ap}j(a;, a)Af’/5(a;, a). (24) Substituting the measured quantities for upB and fu into eqs. (5) and (9) the parameter vector a is usually estimated by minimising some norm of the difference between measured and calculated quantities, called residuals [8] . Using equation (5) is equivalent to the input residual method. Defining the ith partial input residual as (25) where the dependency on the model parameters a of the input vector is defined by f{u), a) := P{uj)F[u), a)uFB(w), (26) the cost function to be minimised is given by M JKa):=£^;}(i)W,(i)^/(i), (27) i=l where Wj{i) represents a weighting matrix for the zth partial residual and the superscript f denotes the conjugate-transpose. The inverse problem (27) is nonlinear with respect to the parameters to be estimated. Thus, there is no advantage relative to the output residual method. Defining the ith partial output residual as vo{i) := u{ui, a) - (28) 1032 where the model output is defined by (29) 0 0 ApiiiuJ^a) . (30) (31) Woii) denotes a weighting matrix for the ith partial output residual. Mathematical modelling is always purpose orientated [9]. In the case discussed in this paper the purpose is to estimate the influence of the foun¬ dation on the dynamics of the rotor. For this purpose, no physically inter¬ pretable model is necessary in order to model this influence. In the next section an alternative mathematical model is introduced which leads to a linear inverse problem. 3.1 ARMA MODELLING IN THE FREQUENCY DOMAIN: THE FILTER MODEL Auto Regressive Moving Average models are well developed (see for in¬ stance [10],[7]) in order to simulate dynamic system behaviour. ARMA models are defined in the time domain by - lAt) = - ^A^), (32) i=0 i=0 where the present output (state or displacement) u{t) due to the present input f{t) depends on rio past outputs and on rii past inputs. In the frequency domain eq. (32) leads to a (frequency-) filter model [7]. With reference to eq. (5) it has the form Su 0 0 0 u(w,a) := [0,0,/2(m+i),0].4 ^(w,a) with the dynamic stiffness matrix A of the entire model A{u, a) = Arii{u) Arib{^^) 0 Arbi{^) Arbb{‘^)B{lo) —B(u) 0 —B{uj) R(a;) 4 -Afbs(^) g) 0 0 Afib{^jO,) the cost function to be minimised is M Jo(a) :=E^oW^oW«o(i). The output and input powers Uo^rii respectively, and the matrices (•SA:)fc=o,-,ni are called filter^parameters and Jiave to be esti¬ mated. Of course the minimum of det[A(a;)] and of det[B(cj)] correspond to the resonance and anti-resonance frequencies of the subsystem of the foundation respectively. For an optimum choice of P (see eq. (9)) the estimation of A and B can be considered to be independent of the precise values of the model of the bearings. Thus, the problem of the uncertainty in the bearmg models and the problem of model estimation are decoupled. If A and B are estimated refering to eq. (5) then = H. (34) The estimation of the filter parameters is robust with respect to deviations in the bearings model. Thus, the uncertainty of the estimation of the foun¬ dation model F is due to the inversion of the matrix P only F = P-^H = p-'-B~^A, (35) which represents a problem a priori and which occurs only in the calculation of the force of the foundation at the bearings /PB = P-^B~^Aufb- (36) Of course the force vector fpB is sensitive to changes in the bearings model but only due to corresponding changes in P. The estimated part B A is robust with respect to changes in the bearings model. In order to calculate the response ufb iio explicit calculation of the inverse of P is necessary, upB = A Bfn- (37) Since the estimated model and the force Jh are insensitive with respect to the bearings model the estimation otupB is robust in this sense. Of course the influence of errors in up b and fu have not yet been taken into account. Accordingly the model powers n^n^, must be estimated as well as the matrices Ai.Bk- The estimation method is outlined in the fol¬ lowing section. 3.2 ESTIMATION OF THE FILTER PARAMETERS In order to estimate the filter parameters the least squares method can be applied to minimise the equation error in eq. (33). Defining the zth partial equation residual as vsij) '= A[ui)upB[^i) - B{uJi)fH{^i) (38) 1034 the cost function to be minimized is given by M i=l (39) where W^ii) denotes a weighting matrix for the ith. equation residual. As¬ suming WeIi) = Am for alH = 1, • • ’, M, the filter equation (33) can be extended for M excitation frequencies as = (40) where C/, 2^ and A are defined by U Z A • • • 5 , r a;i 0 1 lom (41) (42) (43) The solution of the minimisation problem (39) is equivalent to the normal solution of eq. (40) which can be rewritten as [Afio 1 ''' 1 -Ao ) ‘ : ■^o] =: V U Z = 0 (44) Because the filter parameters represented by the matrix V € j^ 4 mx 4 m(no+ni+ 2 ) leal-valued, equation (44) must be satisfied for the real and imaginary parts of the matrix Y € which finally yields 1/[Re {Y} , Im {Y}] =: VA = 0. (45) This problem does not lead to a unique solution for the filter parameters. Indeed, for any arbitrary non-singular matrix C CAufb = (46) is also a solution. But since one is interested (see eq. (34)) in the product B~^ A (or its inverse) only this final result is of interest and this product is unique. 1035 As a necessary and a sufficient condition for a full-rank solution V of eq. (45) the matrix X € 5 ^ 4 m(no+ni 4 - 2 )x 2 M a rank deficiency of 4m, i.e. rank(A) = 4m (no + n^ + 1 ). (47) Of course this problem has to be treated numerically. The rank decision is usually made by looking to the singular values 7 (no, rii) 6 5^4m(no+ni+2) of the matrix X = X{no,ni). Because one cannot expect to achieve zero rather than relative small singular values one has to define a cut-off limit. This is due to the fact that the equation error (39) can be made arbitrary small by increasing the degree p := no + Ui of the filter model. The same situation occurs if one looks at the maximum relative input error 6 / := max (48) or to the maximum relative output error en :== max II^fs(^2)1I (49) With increasing degree p the errors ej and cq can be made arbitrary small. This is a typical expression for an ill-posed problem which can be turned into an well-posed problem by applying regularisation methods [ 11 ]. To choose an appropriate regularisation method needs further investigation and is beyond the scope of this paper. In the next section the method of choosing the optimum force vector fn is demonstrated by a simple example. 4 A SIMPLE EXAMPLE The test model is depicted in Fig. 1. The one-shaft-rotor is simulated by an Euler-Bernoulli beam which is spatially discretised with 10 dof. Accord¬ ing to the partition with respect to inner points and interface points (see eq. ( 3 )) the number n of inner rotor dof is 8 and the number of connecting dof is 2 . Only one translation dof of the rotor is connected to each bearing which are modelled by massless springs with stiffnesses ki — 1.77 • 10® and k 2 = 3-54 • 10® N/m respectively. The foundation is modelled by an uncon¬ nected pair of masses mi = 90, m 2 = 135 kg and springs with stiffnesses kfi = /c /2 = 1.77 • 10® N/m. The force fu due to an unbalance 6 = 0.01 kg-m is given by fu{<^) ■= € IR. The force vector f^i e IR® is assumed to have one non-zero component only, i.e. Jri fuc^. The frequency range between 0 and 250 Hz is discretised with equally spaced stepsize of 0.5 Hz. The selecting matrix T € of the master dof is assumed to consist of the unit vector 64 in order to select out the unbalance force fjj because this 1036 Figure 1: The simple test model excitation is independent of the model of the bearings and therefore of min¬ imum sensitivity. Thus the remaining redundancy consists in eliminating one row of the matrix G := [ei, 82 .83. 65. • • •. eul'' W € (50) The result of the first criterion are depicted in Fig. 2 . It shows the frequency dependent condition numbers for the elimination of each row of the matrix G in turn. This leads to an optimum choice by eliminating the 4-th row of G. Thus, the optimum choice of the master dof is given by T*^ = [ 64 , 65 ]. This result is confirmed by applying the second optimisation criterion. For this purpose a uniform distributed uncorrelated random error with zero mean value is added to the stiffnesses of the bearings simulating a model variation of 50 per cent, i.e. ki —^ ki -1- Nzikil2, where = 1 , 2 , are uncorrelated random numbers with expectation value E{Aki} = 0 and with variance E{AkiAkj} = For a size of I - 500 random samples, the upper and lower bounds /max,/min and /f max,/ h min of the force fpB = /fb(<^, A5) and of the force Jh = /(w,T, AB) respectively with the selecting matrix T - [ 64 , 65 ], have been calculated. In contrast to the maximum difference of upper and lower bounds of the force f^B of ~ 20 , that of the force fn is of the order of the computational accuracy ~ 10 "'^, and is therefore negligible. In a first step the force /fb is used to estimate the filter parameters of the model F of the foundation. Solving the singular value decomposition for all input and output powers {ni.rio) 6 [0, 5]^ C the calculation of the maximum relative input and of the maximum output errors as defined by eqs. (48) and (49) with fn ~ Ifb leads to the results 1037 Elimination of row number... Frequency [Hz] Frequency [Hz] Figure 2: Frequency dependent condition numbers depicted in Fig. 3. For a model realisation with [ni.rio) = (0,2) the values of the maximum relative errors are approximately ej 7.1 • and eo ~ 9.0-10“”^ which corresponds to the computational accuracy. Using this model the estimates of the filter matrices correspond within the computational accuracy to those of the ‘true’ foundation model. Using variations of the force fpB between the bounds /max, /min froni the second criterion the associated upper and lower bounds of the relative input and output error have been calculated. The influence of the variation 1038 Firgure 3: Maximum relative input and output error for different input powers rii and output powers rio using the force vector fpB of the bearing stiffness of 50% leads via the associated variation of the force fpB to drastic variations of the relative input and output errors. The difference of upper and lower bound of the maximum relative input error is 1039 Output power Figure 4: Maximum relative input and output error for different input powers rii and output powers Uo using the optimised force vector Jh of order 100 and that of the maximum relative output error is approximately LOl The situation is different using the optimised force fn in order to es¬ timate the filter parameters of the model H. For each input and output power {ni,no) e [0,25]^ C the maximum relative input error and the maximum relative output error have been calculated. The result is shown in Fig. 4. For a maximum relative input error e; 6.5 • 10“® a filter model of degree 12 is available with the powers (n^, = (4,8). This model produces a maximum relative output error eo ^ 10“®. Analogous to the robustness investigations for fpB now for the force /h the upper and lower bounds of the relative input and of the relative output error due to the random variation in the bearing models have been calculated. For the chosen model with powers (nj,no) = (4,8) the difference of upper and lower bound of the relative input error as well that of the relative output error are of about the same order 10“^. Thus, compared with the order of variation 100 and 10"^ of the direct foundation model estimate the estimate of H is robust with respect to changes in the bearing model. 5 CONCLUSION In this paper a method is introduced which enables the decoupling of the two problems of model estimation and of sensitivity of the foundation model with respect to the model of the bearings. The method produces an opti¬ mised choice of the input/output equation which provides a transfer func¬ tion estimation that is robust with respect to deviations in the model of the bearings. For the foundation model estimation a filter model is intro¬ duced. This modelling strategy has the advantage of leading to a linear inverse problem. The disadvantage is that with increasing model degree the equation error can be made arbitrarily small. Because this error should not become smaller than the accuracy of the data, a cut-off limit has to be determined a priori. Further investigations should allow the cut-off limit to be related to the data errors. REFERENCES 1. Feng, N.S. and Hahn, E.J., Including Foundation Effects on the Vi¬ bration Behaviour of Rotating Machinery. Mechanical Systems and Signal Processing, 1995 Vol. 9, No. 3, pp. 243-256. 2. Friswell, M.I., Lees, A.W. and Smart, M.G., Model Updating Tech¬ niques Applied to Turbo-Generators Mounted on Flexible Founda¬ tions. NAFEMS Second International Conference: Structural Dynam¬ ics Modelling Test, Analysis and Correlation, Glasgow: NAFEMS, 1996 pp. 461-472. 1041 3. Lees, A.W. and Friswell, M.L, Estimation of Forces Exerted on Ma¬ chine Foundations. Identification in Engineering Systems, Wiltshire; The Cromwell Press Ltd., 1996, pp. 793-803. 4. Smart, M.G,, Friswell, M.L, Lees, A.W. and Prells, U., Errors in estimating turbo-generator foundation parameters. In Proceedings ISMA21 - Noise and Vibration Engineering, ed. P. Sas, Katholieke Universiteit Leuven, Belgium, 1996, Vol. II, pp. 1225-1235 5. Zanetta, G.A., Identification Methods in the Dynamics of Turbogener¬ ator Rotors. The International Conference on Vibrations in Rotating Machinery, IMechE, C432/092, 1992, pp. 173-181. 6. Friswell, M.L and Mottershead, J.E., Finite Element Model Updating in Structural Dynamics. Dordrecht, Boston, London: Kluwer Aca¬ demic Publishers, 1995. 7. Natke, H.G., Einfuhrung in die Theorie und Praxis der Zeitreihen- und Modalanalyse - Identifikation schwingungsfdhiger elastomechanis- cher Systeme. Braunschweig, Wiesbaden: Friedrich Vieweg Sz Sohn, 1993. 8. Natke, H.G., Lallement, G., Cottin, N. and Prells, U., Properties of Various Residuals within Updating of Mathematical Models. Inverse Problems in Engineering, Vol. 1, 1995, pp. 329-348. 9. Natke, H.G., What is a true mathematical model? - A discussion of system and model definitions. Inverse Problems in Engineering, 1995, Vol. 1, pp. 267-272. 10. Gawronski, W. and Natke, H.G., On ARMA Models for Vibrating Systems. Probabilistic Engineering Mechanics, 1986, Vol. 1, No. 3, pp. 150-156. 11. Baumeister,J., Stable Solution of Inverse Problems. Braunschweig, Wiesbaden: Friedrich Vieweg &; Sohn, 1987. ACKNOWLEDGEMENTS The authors are indebted to Nuclear Electric Ltd and Magnox Pic for fund¬ ing the research project BB/G/40068/A to develop methods which enables the estimation of the influence of the foundation on the dynamics of the rotor. Dr. Friswell gratefully acknowledges the support of the Engineering and Physical Sciences Research Council through the award of an Advanced Fellowship. 1042 MATHEMATICAL HYSTERESIS MODELS AND THEIR APPLICATION TO NONLINEAR ISOLATION SYSTEMS Y.Q. Ni, J.M. Ko and C.W. Wong Department of Civil and Structural Engineering The Hong Kong Polytechnic University, Hong, Kong Abstract Two mathematical hysteresis models, the Duhem-Madelung (DM) model and the Preisach model, are introduced to represent the hysteretic behavior inherent in nonlinear damping devices. The DM model generates the hysteresis with local memory. Making use of the Duhem operator, the constitutive relation can be described by single¬ valued functions with two variables in transformed state variable spaces. This makes it feasible to apply the force-surface nonparametric identification technique to hysteretic systems. The Preisach model can represent the hysteresis with nonlocal memory. It is particularly suitable for describing the selective-memory hysteresis which appears in some friction-type isolators. An accurate frequency-domain method is developed for analyzing the periodic forced vibration of hysteretic isolation systems defined by these models. A case study of wire-cable vibration isolator is illustrated. 1. Introduction The dynamic response of a structure is highly dependent on the ability of its members and connections to dissipate energy by means of hysteretic behavior. The assessment of this behavior can be done by means of experimental tests and the use of analytical models that take into account the main characteristics of this nonlinear mechanism. Although a variety of hysteresis models have been proposed in the past decades, many structural systems exhibit more complicated hysteretic performance (mainly due to stiffness or/and strength degrading) which the models in existence are reluctant and even inapplicable to depict [1,2]. On the other hand, nonlinear vibration isolation has recently been recognized as one of effective vibration control techniques. In 1043 particular, hysteretic isolation devices have got wide applications owing to their good diy friction damping performance. These hysteretic isolators may exhibit very complicated features such as asymmetric hysteresis, soft-hardening hysteresis, nonlocal selective-memory hysteresis [3-5]. None of the models available currently in structural and mechanical areas can represent all these hysteresis characteristics. Other more elaborate hysteresis models need to be established for this purpose. In reality, hysteresis phenomenon occurs in many different areas of science, and has been attracting the attention of many investigators for a long time. However, the true meaning of hysteresis varies from one area to another due to lack of a stringent mathematical definition of hysteresis. Fortunately, because of the applicative interest and obvious importance of hysteresis phenomenon, Russian mathematicians in 1970’s and the Western mathematicians in 1980’s, began to study hysteresis systematically as a new field of mathematical research [6,7]. They also deal with the hysteresis models proposed by physicists and engineers in various areas, but they separate these models from their physical meanings and formulate them in a purely mathematical form by introducing the concept of hysteresis operators. Such mathematical exposition and treatment can generalize a specific model from a particular area as a general mathematical model which is applicable to the description of hysteresis in other areas. In this paper, two mathematical hysteresis models are introduced and the related problems such as identification and response analysis encountered in their application to nonlinear isolation systems are addressed. 2. Definition of Hysteresis Hysteresis loops give the most direct indication of hysteresis phenomena. But it is intended here to introduce a mathematical definition of hysteresis. Let us consider a constitutive law: u r, which relates an input variable u(t) and an output variable r\t). For a structural or mechanical system, u(t) denotes displacement (strain); r{t) represents restoring force (stress); t is time. We can define hysteresis as a special type of memoiy-based relation 1044 between u(t) and r(t). It appears when the output r[t) is not uniquely determined by the input u(t) at the same instant t, but instead r(t) depends on the evolution of u in the interval [0, t] and possibly also on the initial value ro, i.e. r(t) = iR[u(-),ro](t) (1) where the memory-based functional iR[u(-),ro](t) is referred to as hysteresis operator. In order to exclude viscosity-type memory such as those represented by time convolution, we require that 91 is rate-independent, i.e. that r(t) depends just on the range of u in [0, t] and on the order in which values have been attained, not on its velocity. In reality, memory effects may be not purely rate independent as hysteresis is coupled with viscosity-type effects. However, as shown later, in most cases the rate independent feature of hysteresis is consistent with experimental findings, especially when evolution (variation in time) is not too fast. 3. Duhem-Madelung (DM) Model 3.1 Formulation The DM model can be defined with or without referring to a confined hysteresis region. For the structural or mechanical hysteretic systems, it is not necessary to introduce the notion of bounded curves because there exists neither the saturation state nor the major loop. In this instance, the DM model establishes a mapping (named Duhem operator) 91: (u, ro) rby postulating the following Cauchy problem [7,8] = g, (li.'■) ■ w, (t) - 02(u,r) • u_(t) (2a) r(0) = r„ (2b) where an overdot denotes the derivative with respect to t; gi{u,r] and g 2 {ii,r) are referred to as ascending and descending functions (curves) respectively; and tijt) = max[0,u(t)] = t [|u(t)| + u(t)] (3a) u.(t) = min[0,u(t)] = ^[|u(t)| - u(f)] (3b) Eq. (2a) can be rewritten as f(f) = p[u,r,sgn(u)]-u(t) (4) 1045 u> 0 u<0 (5) in which the describing function has the form g{u,r,sgn{u)] fg{u,r,l) = g,{u,r) [g[u,r-l) = g^(u,r) It is obvious that the DM model is rate independent. In addition, it is specially noted that in this constitutive law the output r{t) is not directly dependent on the entire history of u{t) through [0,t]; but instead depends only on the local histoiy covered since the last change of sgn(u) and on the value of the output at this switching instant. It means that the output can only change its character when the input changes direction. As a consequence, the DM model usually represents the hysteresis with local memory except that the functions gi(u,?) and g2{u,7) are re-specified as hysteresis operators. Within the framework of DM formulation, the ascending and descending functions gi(u,T] and g 2 (u,r) are just required to fulfill suitable regularity conditions and need not to be specified in specific expressions, so both the form and parameters of the functions can be fine-tuned to match experimental findings. On the other hand, the DM formulation can deduce a wide kind of differential-type hysteresis models such as Bouc-Wen model, Ozdemir’s model, Yar-Hammond bilinear model and Dahfs frictional model. For the Bouc-Wen model r(t) = K.u{t) + z(t) (6a) z{t) = au(t] - P|ti(f)|z(f)|z(t)|'”‘ - YU(t)|z(f)|" (6b) it corresponds to the DM model with the specific ascending and descending functions as grj(u,r) = a + K “ [y + p sgn(r -Ku)]|r -Kup (7a) g 2 (w,r) = a + K - [y - p sgn(r -Kw)]|r -ku|” (7b) and for the Yar-Hammond bilinear model f{t) = {a -y sgn(ii) sgn[r - p sgn(ii)]}ii (8) its describing function is independent of u(t) as follows 5f[u,r,sgn(u)l = gf[r,sgn(u)l = a -y sgn(u)sgn[r - p sgn(u)] (9) Hence, the Duhem operator also provides an accessible way to construct novel hysteresis models by prescribing specific 1046 expressions of the ascending and descending functions. Following this approach, it is possible to formulate some models which allow the description of special hysteretic characteristics observed in experiments, such as soft-hardening hysteresis, hardening hysteresis with overlapping loading envelope, and asymmetric hysteresis [5]. 3.2 Identification System identification techniques are classified as parametric and nonparametric procedures. The parametric identification requires that the structure of system model is a priori known. The advantage of nonparametric identification methods is that they do not require a priori the knowledge of system model. The most used nonparametric procedure for nonlinear systems is the force mapping (or called force surface) method [9]. This method is based on the use of polynomial approximation of nonlinear restoring force in terms of two variables—^the displacement u(t) and the velocity u{t ). For nonlinear hysteretic systems, however, the hysteretic restoring force appears as a multivalued function with respect to the variables u{t) and u(t) due to its history- dependent and non-holonomic nature. This renders the force mapping method inapplicable to hysteretic systems, although some efforts have been made to reduce the multivaluedness of the force surface [10,11]. One of the appealing virtues of the DM model is that it can circumvent this difficulty. Making use of the Duhem operator, the hysteretic constitutive relation of Eq.(l) is described by two continuous, single-valued functions gi(u,r) and g 2 {u,r) in terms of the displacement u(t) and the restoring force r{t). Thus, single¬ valued “force” surfaces gi{u,r) and g2{u,7) can be formulated in the subspaces of the state variables {u,r,gi) and (u,r,g 2 ), and can be identified by using the force mapping technique. Following this formulation, a nonparametric identification method is developed by the authors [12]. In this method, the functions g\(u, 7 ) and g2{u,7) are expressed in terms of shifted generalized orthogonal polynomials with respect to u and r as follows gM,r) = i = 0’'(u)G<''0(r) (10a) i=lj=l 1047 (10b) g^r] = z i(|>,.(r)gf(i>,(u) = 0^(u)G'^'<l>(r) I=lj=l where = [gf ]^xn =[5'zf]mxn ^re called the expansion- coefficient matrices of gi{u,r) and g 2 {u,T]. Some algorithms have been proposed to estimate the values of these coefficient matrices based on experimentally observed input and output data. It should be noted that here the vectors 0(u) and <^{r) are shifted generalized orthogonal polynomials [13]. They are formulated on the basis of common recurrence relations and orthogonal rule, and cover all kinds of individual orthogonal polynomials as well as non-orthogonal Taylor series. Consequently, they can obtain specific polynomial-approximation solutions of the same.problem in terms of Chebyshev, Legendre, Laguerre, Jacobi, Hermite and Ultraspherical polynomials and Taylor series as special cases. 4. Preisach Model 4.1 Formulation The intent of introducing the Preisach model is to supply the lack of a suitable hysteresis model in structural and mechanical areas, which is both capable of representing nonlocal hysteresis and mathematically tractable. Experiments revealed that the hysteretic restoring force of some cable-type vibration isolators relates mainly to the peak displacements incurred by them in the past deformation [3]. It will be shown that the Preisach model is especially effective in representing such nonlocal but selective- memory hysteresis, in which only some past input extrema (not the entire input variations) leave their marks upon future states of hysteresis nonlinearities. The Preisach model is constructed as a superposition of a continuous family of elementary rectangular loops, called relay hysteresis operators as shown in Fig. 1. That is [7,14], r(f) ='R[u(-)](f) = j||^(a,P)y„D[u(t)]dadf5 (11) a>P where ia(a,|3) > 0 is a weight function, usually with support on a bounded set in the (a,p)-plane, named Preisach plane; Ya.p[^(^)] is the relay hysteresis operator with thresholds a > p. Outputs of 1048 these simplest hysteresis operators have only two values +1 and -1, so can be interpreted as two-position relays with “up” and “down” positions corresponding to ya,p[^^(^)l=+l and 1 +1 p a -1 Fig. 1 Relay Hysteresis Operator Hence, the Preisach model of Eq.(ll) can be interpreted as a spectral decomposition of the complicated hysteresis operator iR, that usually has nonlocal memory, into the simplest hysteresis operators 7ct,p with local memory. In the following, we illustrate how the model depicts the nonlocal selective-memory feature. Consider a triangle T in the half-plane a > (3 as shown in Fig.2. It is assumed that the weight function )i(a,p) is confined in the triangle T, i.e. }i(a,p) is equal to zero outside T. Following the Preisach formulation, at any time instant t, the triangle T can be subdivided into two sets: S^(t) consisting of points (a,P) for which the corresponding Yc^.p-operators are in the “up” position; and S^(t) consisting of points {a,P) for which the corresponding Ya.p“ operators are in the “down” position. The interface L(t) between S^(t) and Sr(t) is a staircase line whose vertices have a and p coordinates coinciding respectively with local maxima and minima of input at previous instants of time. The nonlocal selective-memory is stored in this way. Thus, the output r(^) a-t any instant t can be expressed equivalently as [14] r{t)= J|^(a,P)dadp- |jM,(a,P)dadp (12) S'(£) S'(t) It should be noted that the Preisach model does not accumulate all past extremum values of input. Some of them can be wiped out by sequent input variations following the wiping-out property {deletion rule): each local input maximum wipes out the 1049 vertices of L(t) whose a-coordinates are below this maximum, and each local minimum wipes out the vertices whose p-coordinates are above this minimum. In other words, only the alternating series of dominant input extrema are stored by the Preisach model; all other input extrema are erased. Fig. 2 Input Sequence and Preisach Plane 4.2 Identification It is seen from Eq.(l 1) that the Preisach model is governed by the weight function |i(a,p) after determining L(t) which depends on the input sequence. !i(a,P) is a single-valued function with respect to two variables a and p. Hence, the aforementioned nonparametric identification method can be also implemented to identify iLL(a,p) by expanding it in a similar expression to Eq.(lO). An alternating approach is to define the following function H'(a',P') = |j|a(a,p)dadp = ^.[^ ^(ot,P)da]dp (13) r(cc'.3') where T(a',p') is the triangle formed by the intersection of the line a = a', p = p' and a = p. Differentiating Eq.(13) yields ^(a',P') = -- aa'ap' (14) Thus, the force mapping identification technique can be applied to determine H(a,p) consistent with the experimental data, and then ^(a,p) is obtained by Eq.(14). 5. Steady-State Response Analysis Hysteretic systems are strongly nonlinear. A study of the steady-state oscillation is one of the classical problems of 1050 nonlinear systems. Usually, the dynamic behavior of a nonlinear system is represented by its resonant frequency and frequency response characteristics. In the following, an accurate frequency- domain method accommodating multiple harmonics is developed to analyze the periodically forced response of hysteretic systems defined by mathematical hysteresis models. Fig.3 shows a single-degree-of-freedom hysteretic oscillator with mass m, viscous damping coefficient c, and linear stiffness ky subjected to an external excitation F(t), for which the governing equation of motion is m • u(t) + c • u(t) + k ' u[t) + r{t) = F(t) (15) where the hysteretic restoring force r(t) is represented by the DM model as Eq.(4). It is worth noting that for the kinetic equation Eq.(15), the excitation is F(t) and the response is u(t); and for the hysteretic constitutive law Eq.(4), u(t) is input and r{t) is output. The causal relationship is different. Fig. 3 Single-Degree-of-Freedom Hysteretic System Due to the hereditary nature of the hysteresis model, it is difficult to directly solve the kinetic equation Eq.(15) by iteration. Here, Eq.(15) is only used to establish the relation between the harmonic components of u(t) and r(t). Suppose that the system is subjected to a general periodic excitation F[t) with known harmonic components F={Fq F^ F 2 ••• F 2 . The multi¬ harmonic steady-state response can be expressed as a ^ ^ . u{t) = ~+ aj cos j(Dt+ aj sin jcot (16) 2 j=i 1051 in which a={ao a 2 ••• a^v ^2 unknown vector containing the harmonic components of u(t). Introducing Eq.(16) into Eq.(15) and using the Galerkin method provide ro^F^-k-a^ (17a) Tj = Fj -c-citj-a] [j = 1,2, ••• , N) (17b) Tj =Fj +c-coj-a^. -(/c-m-coV^)-a* (j = 1, 2, ,N) (17c) where r={ro q ••• r^}'^ is the harmonic vector of the hysteretic restoring force r(t). Referring to the hysteretic constitutive law, we define the determining equation as D(t) = r{t) - g[u,r,sgn(u)] • u{t) (18) When a is the solution of u(t), applying the Galerkin method into Eq.(18) and considering Eq.(17) achieve d{a) = 0 (19) where the vector d{a)={dQ ^3 --dj^ d^ d^ "• is comprised of the harmonic components of D(t) corresponding to a. An efficient procedure to seek the solution of Eq.(19) is the Levenberg-Marquardt algorithm with the iteration formula where the Jacobian matrix J[a(^)] = dd(a)/da\a=a^^) ; 9 ^ is the Levenberg-Marquardt parameter and I is identity matrix. At each iteration, the function vector and Jacobian matrix should be recalculated with updated values of Here, a frequency/time domain alternation scheme by FFT is introduced to evaluate the values of d(a) and J[a) at d{a) and dd[a)/da are known to be the Fourier expansion coefficients of D(t) and dD(t)/ da respectively. For a given a(^) and known F, the corresponding r[a(^)] is obtained from Eq.(17), and the inverse FFT is implemented for and r[a(^)] to obtain all the time domain discrete values of u{t], u(t) , r(t) and f(t) over an integral period. Then the time domain discrete values of the function D{u,u,r,r,t), corresponding to a=a(^), are evaluated from Eq.(18). Making use of forward FFT to these time domain discrete values of D{u,u,r,r,t) , the values of function vector d[a(^)] are obtained. 1052 Similarly, the partial differential dD{t) / da can be analytically evaluated in the time domain. Forward FFT to the time domain values of dD(t) / da at ct=a(^) gives rise to dd[ai^)]/da. 6. Case Study Wire-cable vibration isolators are typical hysteretic damping devices. Dynamic tests show that their hysteresis behaviors are almost independent of the frequency in the tested frequency range [4,15]. Experimental study and parametric modelling of a wire-cable isolator have been carried out [5]. Fig.4 shows the experimental hysteresis loops in shear mode. It is seen that for relatively small deformations, the isolator exhibits softening hysteresis loops. When large displacements are imposed, the stiffness of the loops becomes smoothly hard. This nature is referred to as soft-hardening hysteresis. Based on the Bouc-Wen model, a parametric identification was performed to model these hysteresis loops, but the result is unsatisfactory. This is due to the fact that the Bouc-Wen model cannot represent such soft- hardening nature of hysteresis. Fig. 4 Experimental Hysteresis Loops We now use the DM model to represent these hysteresis loops, and perform a nonparametric identification to determine the functions gi(u,7) and g2(u,7). The simplest Taylor series are adopted, i.e. (i),(r) = (r / and (|)j(u) = (u j UqY~^ (ro = 20.0 and uo = 2.0). Fig.5 shows the identified “force” surfaces of gi{u,T) and g 2 (u,r) by taking m=n~S. Fig.6 presents the theoretical hysteresis 1053 loops generated by the DM model using the identified g\(u,T) and g 2 (u,}). It is seen that the modeled hysteresis loops are agreeable to the observed loops. In particular, the soft-hardening nature is reflected in the modeled hysteresis loops. -200 Fig. 6 Modeled Hysteresis Loops by DM Model After performing the modelling of hysteretic behavior, the dynamic responses of hysteretic systems can be predicted by the developed method. Fig.7 shows a vibration isolation system installed with wire-cable isolators in shear mode. It is subjected to harmonic ground acceleration excitation Xg[t)=Acos2Kft. The equation of motion of the system is expressed as m-u(t) +K ■r(t) =-rri'Xg(t) (21) where m is the mass of the system; K is number of the isolators installed. u{t) is the displacement of the system relative to the ground. r(t) is the restoring force of each isolator and has been determined from nonparametric identification. Fig.8 illustrates the predicted frequency-response curves of the relative displacement when 7n=6kg and K=2. The excitation amplitude A is taken as 0.25g, O.SOg, 0.35g, 0.40g and 0.45g respectively. The frequency-response curves show clearly the nonlinear nature of the wire-cable isolation system. i I Xg(t) Fig. 7 Vibration Isolation System with Wire-Cable Isolators Fig. 8 Frequency Response Curves of Relative Displacement 1055 7. Concluding Remarks This paper reports a preliminaiy work of introducing the mathematical hysteresis models in structural and mechanical areas. It is shown that a wide kind of differential hysteresis models, which are extensively used at present, can be derived from the Duhem-Madelung (DM) model. Thus, the mathematical properties concerning the DM model are also possessed by these models. Two potential advantages appear when the DM formulation is used. Firstly, it allows to apply the force mapping technique to hysteretic systems. Secondly, it provides an approach to construct novel differential models which reflect some special hysteretic characteristics. The Preisach model is shown to be capable of representing nonlocal hysteresis and mathematically tractable. It offers a more accurate description of several observed hysteretic phenomena. Emphasis is placed on demonstrating the selective-memory nature of this hysteresis model. The case study based on experimental data of a wire- cable isolator has shown the applicability of the mathematical hysteresis model, and the validity of the steady-state response analysis method proposed in the present paper. Acknowledgment: This study was supported in part by the Hong Kong Research Grants Council (RGC) and partly by The Hong Kong Polytechnic University. These supports are gratefully acknowledged. References 1. Azevedo, J. and Calado, L., ‘‘Hysteretic behaviour of steel members: analytical models and experimental tests”, J. Construct Steel Research, 1994, 29, 71-94. 2. Kayvani, K. and Barzegar, F., “Hysteretic modelling of tubular members and offshore platforms”, Eng. Struct, 1996, 18, 93- 101 . 3. Lo, H.R., Hammond, J.K. and Sainsbury, M.G., “Nonlinear system identification and modelling with application to an isolator with hysteresis”, Proc. 6th Int modal Anal Conf., Kissimmee, Florida, 1988, Vol.II, 1453-1459. 1056 4. Demetriades, G.F., Constantinou, M.C. and Reinhorn, A.M., “Study of wire rope systems for seismic protection of equipment in buildings”, Eng. Struct, 1993, 15, 321-334. 5. Ni, Y.Q., “Dynamic response and system identification of nonlinear hysteretic systems”, PhD Dissertation, The Hong Kong Polytechnic University, Hong, Kong, November 1996. 6. Krasnosefskii, M.A. and Pokrovskii, A.V., Systems with Hysteresis, translated from the Russian by M. Niezgodka, Springer-Verlag, Berlin, 1989. 7. Visintin, A., Differential Models of Hysteresis, Springer-Verlag, Berlin, 1994, 8. Macki, J.W., Nistri, P. and Zecca, P., “Mathematical models for hysteresis”, SIAM Review, 1993, 35, 94-123. 9. Masri, S.F. and Caughey, T.K., “A nonparametric identification technique for nonlinear dynamic problems”, J. Appl Mech, ASME, 1979, 46, 433-447. 10. Lo, H.R. and Hammond, J.K., “Nonlinear system identification using the surface of nonlinearity form: discussion on parameter estimation and some related problems”, Proc. 3rd Int Conf Recent Adv. Struct. Dyn., Southampton, UK, 1988, 339-348. 11. Benedettini, F., Capecchi, D. and Vestroni, F., “Identification of hysteretic oscillators under earthquake loading by nonparametric models”, J. Eng. Mech., ASCE, 1995, 121, 606- 612. 12. Ni, Y.Q., Ko, J.M. and Wong, C.W., “Modelling and identification of nonlinear hysteretic vibration isolators”. Accepted to SPJE’s 4th Annual Symposium on Smart Structures and Materials : Passive Damping and Isolation, 3-6 March 1997, San Diego, USA. 13. Ni, Y.Q., Wong, C.W. and Ko, J.M., “The generalized orthogonal polynomial (GOP) method for the stability analysis of periodic systems”, Proc. Int. Conf. Comput. Methods Struct. Geotech. Eng., Hong Kong, 1994, Vol.II, 464-469. 14. Mayergoyz, I.D., Mathematical Models of Hysteresis, Springer- Verlag, New York, 1991. 15. Ko, J.M., Ni, Y.Q. and Tian, Q.L., “Hysteretic behavior and empirical modeling of a wire-cable vibration isolator”, Int. J. Anal. Exp. Modal Anal, 1992, 7, 111-127. 1057 1058 The identification of turbogenerator foundation models from run-down data M Smart, M I Friswell, A W Lees, U Prells Department of Mechanical Engineering University of Wales Swansea, Swansea SA2 8PP UK email: m.smart@swansea.ac.nk ABSTRACT The trend of placing turbines in modern power stations on flexi¬ ble steel foundations means that the foundations exert a considerable influence on the dynamics of the system. In general, the complex¬ ity of the foundations means that models are not available a priori, but rather need to be identified. One way of doing this is to use the measured responses of the foundation at the bearings to the synchronous excitation obtained when the rotor is run down. This paper discusses the implementation of such an estimation technique, based on an accurate model of the rotor and state of unbalance, and some knowledge of the dynamics of the bearings. The effect of errors in the bearing model and response measurements on the identified parameters is considered, and the instrumental variable method is suggested as one means of correcting them. 1 INTRODUCTION The cost of failure of a typical turbine in a modern power station is very high, and therefore development of condition monitoring techniques for such machines is an active area of research. Condition monitoring relies on measuring machine vibrations and using them to locate and quantify faults, which obviously requires an accurate dynamic model of the ma¬ chine. Although the dynamic characteristics of rotors are generally well understood, the foundations on which they rest are not. Since the founda¬ tions are often quite flexible, they can contribute considerably to the rotor’s dynamic behaviour. Finite element modelling has been attempted but the complexity of the foundations, and the fact that they often differ substantially from the original drawings rendered the technique generally unsuccessful[1]. Exper¬ imental modal analysis is another possible solution, but this requires that the rotor be removed from the foundation, and that all casings remain in place, which is not practical for existing power plant. However, mainte¬ nance procedures require that rotors are run down at regular intervals and this procedure provides forcing to the foundation over the frequency range 1059 of operation. By measuring the response at the bearing pedestals (which is already performed for condition monitoring purposes) an input-output relation for the foundation may be obtained. Lees [2] developed a least-squares method to calculate the foundation parameters by assuming that an accurate model exists for the rotor, that the state of unbalance is known from balancing runs, and that the dynamic stiffness matrices of the bearings can be calculated. Although bearing mod¬ els are not in fact well characterised, Lees and Friswell[3] showed that the parameter estimates are only sensitive to the bearing stiffnesses over limited frequency ranges, which can be calculated. Feng and Hahn[4] followed a similar approach but added extra informa¬ tion by measuring the displacements of the shaft. Zanetta[5] also measured the shaft displacements but included the bearing characteristics as param¬ eters to be estimated. Although any extra information is desirable in a parameter estimation routine the equipment necessary to measure these quantities only exists in the newer power stations, and it was desired to make the method applicable to older plant as well. In the analysis presented here, the measured data consists of the motion of the bearing pedestals in the horizontal and vertical directions, although not necessarily in both di¬ rections at every bearing. 2 THEORY 2.1 Force estimation If D is the dynamic stiffness matrix of a structure defined as D{u) K ^luC -oj^M (1) where M,C,K are the mass, damping and stiffness matrices then Dx = f (2) where x is response and f force. Referring to figure 1 it is seen that the rotor is connected to the foundation via the bearings. It is assumed that good models exist for both rotor and bearings, and that the state of unbalance is known. The implications of these assumptions will be discussed later. The dynamic stiffness equation for the whole system may be written as The subscripts r and / refer to the rotor and foundation degrees-of-freedom respectively, u refers to the unbalance forces and b to the bearing forces. 1060 ROTOR Figure 1: Rotor-bearing system There is a negative sign before the bearing forces ft, since they refer to the forces acting on the bearings. The foundation d.o.f are those where the re¬ sponses are measured, in other words no internal d.o.f are represented. Df therefore represents a reduced dynamic stiffness matrix. The response mea¬ surements will not be the total vibration level at the bearings but rather the vibrations at once-per-revolution and it is assumed that no dynamic forces at this frequency will be transmitted to the foundation via the substructure onto which it is fixed. The equation for the bearings in the global coordinate system is / Dbrr Dbrf W ^7- ^ ^ fbT\ V Dij, Dill )\^s) \fbs) (4) This assumes that the bearings behave as complex springs, in other words they have negligible inertia and no internal d.o.f. Substituting (4) into (3) we have / Dt„ Dtrl \ ( ^'■ \ = ( V A/r A// J [ J A ° / where / Dtrl 0 \ f Dirr Di^l \\ [Dtlr DtllJ-[[ 0 DlJ + [Diir Dill)) (5) ( 6 ) 1061 and where subscript t refers to the total model. Solving equation (5) for Xr leads to Xr — J^trr ifur ~~ and solving equation (4) for ft / yields fbf — ^fhr ~ ~^brr^r ~ ^brf^f (S) All quantities in equation (8) are known either from assumed models {Dr, D^) or experiment {xf, /^r)- This calculated force fbf may then be used to¬ gether with the measured responses to estimate the foundation parameters. 2.2 Foundation parameter estimation Once the forces have been estimated, the foundation parameters must be derived. The dynamic stiffness equation for the foundation is DfXf — fbf (9) Although D/ is a reduced stiffness matrix it is assumed that it has the form of equation (1). Therefore equation (9) may be written as W{u)v = fbf (10) where is a column vector formed from the elements of M, C and K and W is a matrix formed from the response vector which depends explicitly on u. Clearly this is an under-determined set of equations, but by taking mea¬ surements at many frequencies it may be made over-determined, and thus solvable in a least squares sense. Since the magnitude of the mass, damp¬ ing and stiffness elements normally differ by several orders-of-magnitude, it was found expedient to scale the mass parameters by o;^, and the damping parameters by uJ, where u is the mean value of the frequency. 2.3 Errors in estimates It is necessary now to examine the effect of errors on the parameter esti¬ mates. Equation (10) is of the form Ax = b, where A has dimension mxn. In this particular case, A depends on the measured response Xf, whilst b depends on the measured response, applied unbalance, and assumed rotor and bearing models. Therefore the estimated parameters will be sensitive to the following errors: 1. Errors in the rotor model 2. Errors in the bearing model 3. Errors in the state of unbalance 1062 4. Errors in the measured foundation response The rotor model is generally well known, as is the state of unbalance, so the main source of error in the estimates is due to measurement noise and bearing uncertainty. If the least squares problem is formulated as Ax = b (11) then (Aq + Ajv)ic = bo + b/i/ (12) where the subscript N refers to noise and 0 to data which is noise-free. The least-squares estimate is given by X = (Aq Ao + A^Aiv + + A^Ao) ^(Aq + Ayv)^(bo + b;v) (13) Even if the noise on the outputs is uncorrelated with the noise on the inputs the expected value of x does not equal that of its estimate: E[x] - E[x] 0 (14) In other words the estimate x is biased [6]. In order to reduce the bias of the estimates, the instrumental variables method can be used. Essentially, it requires the use of a matrix that is uncorrelated with the noise on the outputs, but which is strongly correlated with the noise-free measurements themselves. If W is the instrumental variable matrix, then W'^Ax = W'^b (15) Expanding i = (W^Ao + + b^) (16) This means that E[x] = x, in other words unbiased estimates result. Fritzen[7] suggested an iterative method for solving for the parameters. Initially, equation (10) is solved in a least-squares sense, and the values of the estimated parameters are used to calculate outputs for the model. These outputs are then used to create W in the same way as the original outputs were used to create A, new estimates are obtained, and if neces¬ sary the process is repeated. Experience seems to suggest good convergence properties[7]. 3 SIMULATION The method under discussion in this paper was tested on a model of a small test rig located at Aston University, Birmingham. This consists of a steel shaft approximately 1.1m long, with nominal diameter 38mm. The shaft is supported at either end by a journal bearing of diameter 100mm, 1063 1 2 3 4 5 6 7 8 9 10 11 12 Figure 2: Rotor-bearing system L/D ratio of 0.3 and clearance of 25/.tm. There are two shrink-fitted bal¬ ancing discs for balancing runs. Each bearing is supported on a flexible pedestal to simulate the flexible foundations encountered in power station turbines. At present these pedestals are bolted onto a massive lathe bed. The rotor is powered by a DC motor attached via a belt to a driving pulley, which is in turn attached via a flexible coupling to the main rotor shaft. A schematic of the rig is shown in figure 2. Dimensions of each station and material properties are given in table 1. A finite element model of the rotor with 23 elements was created and short bearing theory was used to obtain values for the bearing stiffness and damping[8]. The pedestals themselves consist of two rectangular steel plates, 600mm X 150mm which have two channels cut into them, and which are supported on knife-edges (figure 3). The vertical stiffness arises from the hinge effect of the channels, whilst the horizontal stiffness is as a result of the shaft centre tilting under an applied load. Treating the supports as beams, the theoretical stiffnesses are: Ky = 0.^5MN/m !<:, = 1.5MiV/m where x and y refer to the horizontal and vertical directions respectively. The masses and damping factors were taken as: il/4 = My = 50/cp Ca: = Cy = 150iV • s/m The estimation theory was tested using this model. The finite element model was used to generate responses at the bearings for frequencies from 1064 Table 1: Table of rotor rig properties Shaft Properties Station Length (mm) Diameter (mm) E (GPa) P (kg/m^) 1 6.35 38.1 200 7850 2 25.4 77.57 200 7850 3 50.8 38.1 200 7850 4 203.2 100 200 7850 5 177.8 38.1 200 7850 6 50.8 116.8 200 7850 7 76.2 38.1 200 7850 8 76.2 109.7 200 7850 9 76.2 38.1 200 7850 10 50.8 102.9 200 7850 11 177.8 38.1 200 7850 12 203.2 100 200 7850 Balancing discs Station Length (mm) Diameter (mm) Unbalance (kg • m) 6 25.4 203.2 0.001 25.4 203.2 0.001 0 to 30 Hz with a spacing of 0.1 Hz. The responses were corrupted by normally distributed random noise with zero mean and standard deviation of 0.1% of the maximum response amplitude (applied to both real and imaginary parts of the response). At each frequency the bearing static forces were disturbed by noise drawn from a uniform distribution spanning an interval of 20% of the force magnitude, to introduce uncertainty into the bearing parameters. The unbalance was assumed to be exactly known. A series of 30 runs was performed, foundation parameter estimates calculated and the mean and standard deviations of these estimates obtained. The magnitudes of the responses at both bearings are given in figure 4, which show that there are four critical speeds in the frequency range under consideration. A sample of true and estimated forces in the bearings are shown in figures 5 and 6. The means {fi) and standard deviations (a) of the least-square (LS) and instrumental variable (IV) estimates for the foundation parameters are shown in table 2. Displacement (m) Figure 3: Flexible bearing supports Figure 4: Magnitudes of responses at bearings 1066 Table 2; Parameter estimates for foundations showing uncertainty 1069 4 DISCUSSION The results in table 2 show a clear improvement in parameter estimates when the instrumental variable method is used. There is a clear bias in the least-squares estimates which is significantly less when the instrumen¬ tal variable method is employed. Also, despite the fact that the bearing parameters are assumed to be seriously in error, the estimates appear to be insensitive to them. This will be true provided that the bearings are much stiffer than the foundation (a reasonable assumption in practice). It does appear however that in some cases the standard deviation of the instru¬ mental variable estimate is larger than that of the least-squares estimate, a fact which warrants further investigation. As far as the rotor model is concerned, impact tests, which are per¬ formed on rotors prior to them entering service, normally give experimental frequencies which are within a few percent of the theoretical ones. Thus the assumption that the rotor model is accurately known would appear to be reasonable. The state of unbalance may in theory be established from a balancing run. If two successive run-downs are performed, one due to the unknown system unbalance and one with known balance weights attached, then pro¬ vided the system is linear the response measurements may be vectorially subtracted to give the response due to the known balance weights alone. In order to ascertain the effect of unbalance uncertainty on the parameter estimates, one run was performed assuming no error in the unbalance. It should be noted that this assumes that the system is linear. 5 CONCLUSIONS A method of estimating turbogenerator foundation parameters from potentially noisy measurement data is demonstrated. It is shown that making use of the instrumental variable method reduces the bias in the estimates and improves them quite significantly. 6 ACKNOWLEDGEMENT The authors wish to acknowledge the support and funding of Nuclear Electric Ltd and Magnox Electric Pic. Dr Friswell wishes to acknowledge the support of the Engineering and Physical Sciences Research Council through the award of an advanced fellowship. 1070 REFERENCES [1] A. W. Lees and I. C. Simpson. The dynamics of turbo-alternator foun¬ dations: Paper C6/83. In Conference on steam and gas turbine founda¬ tions and shaft alignment, Bury St Edmunds, 1983, IMechE, pp37-44. [2] A. W. Lees. The least squares method applied to identify ro¬ tor/foundation parameters: Paper C306/88. In Proceedings of the Inter¬ national Conference on Vibrations in Rotating Machinery, Edinburgh, 1988, IMechE, pp209-216. [3] M. 1. Friswell and A. W. Lees. Estimation of forces exerted on machine foundations. In M. 1. Friswell and J. E. Mottershead, editors, Interna¬ tional Conference on Identification in Engineering Systems, Swansea, 1996, pp793-803. [4] N. S. Feng and E. J. Hahn. Including foundation effects on the vibra¬ tion behaviour of rotating machinery. Mechanical Systems and Signal Processing, 1995, 9, pp243-256. [5] G. A. Zanetta. Identification methods in the dynamics of turbogener¬ ator rotors: Paper C432/092. In IMechE Conference on Vibrations in rotating machinery, Bath, 1992. IMechE, ppl73-181. [6] J. Schoukens and R. Pintelon. Identification of linear systems. Perga- mon Press, 1991. [7] C. P. Fritzen. Identification of mass, damping and stiffness matrices of mechanical systems. Journal of Vibration, Acoustics, Stress and Relia¬ bility if Design, 1986 108, pp9-17. [8] D. M. Smith. Journal bearings in Turbomachinery. Chapman and Hail, 1969. 1071 1072 SHELL MODE NOISE IN RECIPROCATING REFRIGERATION COMPRESSORS Ciineyt Oztiirk and Aydin Bahadir Tiirk Elektrik Endiistrisi A.$ R&D Department Davutpa§a, Litres Yolu, Topkapi -34020, Istanbul, Turkey ABSTRACT This study describes the successful endeavor to understand the causes of noise that appear on the shell modes of the reciprocating refrigeration compressors. The compressor shell is generally considered as the acoustic enclosure that reflects the acoustic energy back into the compressor cavity but also as the transmitter and radiator of the transmitted acoustic energy that could be radiated into the air or transmitted to the structure. Vibrations of the compressor shell can easily be characterized in terms of the modal parameters that consist of the natural frequencies, mode shapes and damping coefficients. The noise source harmonics and the shell resonances couple to produce the shell noise and vibration. The harmonic spacing is equal to the basic pumping frequency. Results of the studies indicate that important natural frequencies of the compressor shell usually stay between 2000-6000 Hz interval. The important natural frequencies are first natural frequencies in the lower range with the longer wavelengths that radiate well. INTRODUCTION Compressor noise sources are those processes where certain portions are separated from the desired energy flow and transmitted through the internal components of compressor to the hermetic shell where it is radiated from the shell as airborne noise on vibration of supporting structure will eventually radiate noise from some portion of the structure. Noise sources of the reciprocating refrigeration compressors can be classified as motor noise, compression process noise and valve port flow noise. 1073 In reciprocating compressors there is very high density of noise harmonics even though they decay in amplitude at high frequencies. Generally, these noise source harmonics and the shell resonances couple to produce shell noise and vibration NOISE GENERATION MECHANISM OF THE RECIPROCATING REFRIGERATION COMPRESSORS Significance of the problem The results of the sound radiation characteristics shown at figure 1 indicated that certain high amplitude frequency components had very distinctive sound radiation patterns. It was suspected these frequencies correspond to excitation of either structural resonances of the compressor shell or acoustic resonances of the interior cavity space. Resonances those amplify the noise and vibration caused by pumping harmonics of a compressor and thus can be the cause of significant noise problems. Sound power - A-Wcighted —I-,-1-,-;-P——r— f —!——'I' "I '"'I —I- ”r ' i i 50.0 63.0 80,0 100,0 125,0 160,0 200,0 250,0 315,0 400,0 500,0 630,0 800,0 l,0k t,3k 1,6k 2,0k 2,5k 4,0k S,0k 63k S,0k 10,0k Frequency [Hz] Figure 1, Noise Radiation Characteristics of Reciprocating Refrigeration Compressor. Noise sources Noise in a compressor is generated during cyclic compression, discharge, expansion and suction process. The character of noise sources is harmonic due to periodic nature of the compression process. These harmonics are present in the compression chamber, pressure time history and loading of the compressor through drive system. The motor can not provide immediate response to load harmonics and load balance is obtained at the expense of acceleration and 1074 deceleration of the motor drive system. Harmonic vibrations of the motor drive system can then excite the resonant response in the compressor components that can transmit the acoustic energy in very efficient way. The rest of the noise sources are, turbulent nature of flow depending to passage through valve ports, valve impacts on their seats and possible amplification when matched with mechanical resonances. MECHANICAL FEATURES OF SHELL Mechanical features of the compressor The hermetically sealed motor compressor comprises in general a motor compressor unit including a motor assembly mounted with a frame and a sealed housing within which the compressor is supported by means of plurality of coil springs each having one end spring with the frame and the other end connected with the interior of the housing. The refrigerant gas as it is compressed in the cylinder is discharged through the discharge chamber in the cylinder head into the discharge muffler. The discharge muffler is generally mounted on the cylinder head attached in covering relation to an end face of the cylinder. Where the sealed casing is spherical in shape for better noise suppression, an upper end of the cylinder head tends to interfere with an inner wall surface of the casing, a disadvantage that can only be eliminated by increasing the size of scaled casing for providing a desired hermetically sealed motor. Compressor Shell The shell is easily be characterized with the modal parameters. The ideal shell should be designed in a way that keep all the excitation frequencies at the mass controlled region of all its modes. But, depending on the very tight constraints that come with the gas dynamics and motor locations, it is not allowed to be flexible during the design of shell. As a consequence of the existing design limitations natural frequencies of the shell usually fall between 1000-5000 Hz. SHELL RADIATION Figure 1 illustrates how the sound pressure level of a pumping harmonic can increase as it nears a resonant frequency. The sound pressure level of the pumping harmonic increases around the shell resonances. The resonance generally radiates primarily from the large flat sides of the compressor. There are three major acoustic cavity resonances 400, 500 and 630 Hz and four major structural resonances of the compressor shell: around 1.6 K, 2 K, 2.5 and 3.2 K. Hz at which noise radiates well in certain directions. 1075 SHELL RESONANCES Figures 2, 3 and 4 illustrate the frequency responses of the compressor shell when measured with the impact hammer method. Figure 2 is the response to the excitation in x direction, figure 3 is for the excitation direction and figure 4 is for the excitation in z direction. Figure 2, Frequency response of the compressor shell when excited in X direction. Figure 3, Frequency response of the compressor shell when excited in Y direction. 1076 Figure 4, Frequency response of the compressor shell when excited in Z direction. To verify the hypothesis that resonances were contributing to some of the noise problems of the reciprocating piston compressor, a modal analysis of the shell and interior cavity was performed. For the shell modal analysis, the accelerometer to measure the response remained stationary, while the impact location was moved. The test was performed in this manner for convenience since it was easier to fix the accelerometer in one location and strike the compressor with force hammer at each grid point to obtain transfer function for each measurement location. Identical results are obtained if impact occurred at a single point and the response was measured at each measurement location. Preliminary test were performed initially to identify an appropriate measurement location at which all important natural frequencies of shell are detectable. Several force input and response locations were evaluated to determine the best location to mount accelerometer to measure the shell response. The shell resonances are also calculated by using the Structural FEM analysis. Table-1 lists the natural frequencies predicted in these studies. During the finite element analysis, the models of the compressor were built, based on the CAD models. The shell consists of 7500 elements. The mesh densities are quite adequate for the structural analysis in the frequency range of interest. In order to investigate the possible influence of the crank mechanism on the natural frequencies of the shell, a simple model of the crank mechanism was introduced to the FE model. During the calculations, the crank mechanism was simplified as a rigid block with certain mass and rotary inertia and modeled with solid elements. The shell and the crank mechanism have been suspended with the 1077 springs from 3 positions and in all 6 transitional and rotational directions. Depending on the negligible spring effect on the longitudinal direction, estimated values have been used in 5 directions. The FE model has been assumed to be free-free. Mode # Frequency Hz Mode # Frequency Hz 1 1997 21 4716 2 22 4925 3 2293 23 4994 4 24 5 25 5159 6 26 5454 7 2889 27 5476 8 3258 28 9 29 EESI^HI 10 30 5783 11 3376 31 5936 12 3551 32 5999 13 3577 33 6035 14 34 6055 15 3788 35 6183 16 3958 36 6237 17 4383 37 6314 18 4481 38 19 4644 39 20 4702 40 6701 1 Table-1, Calculated Natural Frequencies of the Shell Figures 5, 6, 7 and 8 illustrate how the mode shapes of the shell vary at the mode frequencies of 2754, 3332, 3551 and 3788 Hz , These figures indicate that the shell vibrates predominantly along the large flat sides of the compressor at points where the suspension springs are attached to the shell wall at these natural frequencies. When referred to figure 1 of the noise radiation this frequency range is also the range where the noise radiates efficiently from the large flat sides of the compressor. Thus, there is good correlation between the acoustic data and structural data for these frequencies. The slight discrepancies in the structural natural frequencies and the acoustic data. Acoustic data have been picked up at the shell temperature of the reciprocating piston compressor that could reach up to 110 C. The modal analysis results also indicate that the compressor suspension springs are attached to a point on the shell where the shell is comparatively compliant. Thus, the vibrational energy transmitted through the springs to the compressor shell can and did effectively excite the shell vibrations. Also, significant shell vibrations occur along the large flat sides of the compressor shell indicating the curvature of the shell needs to be increased to add stifihiess to the shell. Based on the results of the shell modal analysis, it is recommended the suspension springs moved away from the compliant side walls of the shell. A four spring arrangement at the bottom of the shell near corners where the curvature is sharp would reduce the amount of vibration energy transferred to the shell because of the reduced input mobility of the shell at these locations. It is also believed increasing the stiffness of the shell by increasing the curvature will provide noise reduction benefits. The greater shell stiffness lowers the amplitude of the shell vibrations. Figure 9, illustrate the third octave change in compressor noise with the same compressor in the new shell. An over all noise level of 5 dBA has been obtained. Figure 9 Compressor noise level improvement after the shell modification. The increased shell stiffness also raises the natural frequencies of the shell where there is less energy for transfer function response. However, there is a possible disadvantage to increasing stiffness of the shell. The higher natural frequency lowers critical frequency of the shell thus reducing transmission loss of the shell. 1081 Damping treatments can also have obvious benefits in vibration and noise reduction. Visco elastic and Acoustic dampings are considered to avoid the shell excitations. The application of dampers can also provide up to 5 dBA reductions when appropriately located on the shell. CONCLUSION The results of studies indicate that structural resonances of the shell are indeed amplifying the noise due to the pumping harmonics of the reciprocating refrigeration compressor to cause significant noise radiation outside of compressor. In order to tackle with this noise problem, within the scope of these studies two different effective shell noise control are considered based on the results of numerical and experimental structural analysis and acoustic features of the reciprocating compressor. First, shell noise control method is the redesign of the shell with increased stiffness by replacing all the abrupt changes in the curvature with the smooth continuous changes. It is apparent that change in the shell configuration can shift the first shell resonance from 1750 Hz to nearly 3200 Hz. The results of the redesign effects can reach up to 3-5 dBA reduction on third octave noise levels. Second treatment that could be applied against the excitation of shell resonances are considered as the acoustic and viscoelastic dampers. These dampers can be chosen to operate efficiently at the shell resonant frequencies. These two applications can also provide up to 2-5 dBA reduction on the third octave band of the noise emission but the long term endurance and temperature dependence of these materials can always be a question mark when considered from the manufacturer point of view. REFERENCES 1. JFROBATTAand ID. JONES 1991, Purdue University, School of Mechanical Engineering, The Ray Herrick Laboratories, Report no: 1912-1 HL 91-9P, 73-84, Investigation of Noise Generation Mechanisms and Transmission Paths of Fractional Horsepower Reciprocating Piston and Rolling Piston Compressors 2. HAMILTON IF 1988, Purdue University, School of Mechanical Engineering, The Ray Herrick Laboratories, 207-213 Measurement and Control of Compressor Noise 3. C OZTURK, A AQIKGOZ and J L MIGEOT 1996, International Compressor Engineering Conference at Purdue, Conference Proeceeding, Volume II, 697-703, Radiation Analysis of the Reciprocating Refrigeration Compressor Casing 1082 A COMPARATIVE STUDY OF MOVING FORCE IDENTIFICATION T.H.T. Chan, S.S. Law, T.H. Yung Department of Civil & Structural Engineering, The Hong Kong Polytechnic University, Hung Horn, Kowloon, Hong Kong ABSTRACT Traditional ways to acquire truck axle and gross weight information are expensive and subject to bias, and this has led to the development of Weigh- in-Motion (WIM) techniques. Most of the existing WIM systems have been developed to measure only the static axle loads. However dynamic axle loads are also important. Some systems use instrumented vehicles to measure dynamic axle loads, but are subject to bias. These all prompt the need to develop a system to measure the dynamic axle loads using an unbiased random sample of vehicles. This paper aims to introduce four methods in determining such dynamic forces from bridge responses. The four methods are compared with one another based on maximum number of forces to be identified, minimum number of sensors, sensitivity towards noise and the computation time. It is concluded that acceptable estimates could be obtained by all the four methods. Further work includes merging the four methods into a Moving Force Identification System (MFIS). INTRODUCTION The truck axle and gross weight information have application in areas such as the structural and maintenance requirements of bridges and pavements. However, the traditional ways to acquire that are expensive and subject to bias, and this has led to the development of Weigh-in-Motion (WIM) techniques. Some systems are road-surface systems which make use of piezo-electric (pressure electricity) or capacitive properties to develop a plastic mat or capacitive sensors to measure axle weight [1]. Another kind of WIM system is the under-structure systems in which sensors are installed under a bridge or a culvert and the axle loads are computed from the measured responses e.g. AXWAY [2] and CULWAY [3]. All the above mentioned systems can only give the equivalent static axle loads. However dynamic axle loads are also important as they may increase road surface damage by a factor of 2 to 4 over that caused by static loads [4]. Some systems use instrumented vehicles to measure dynamic axle loads [5], but are subject to bias. These all prompt the need to develop a system to measure the dynamic axle loads using unbiased random samples of vehicles. Four methods are developed to determine such 1083 dynamic forces from bridge responses which include bending moments or accelerations. EQUATION OF MOTIONS FOR MOVING LOADS The moving force identification methods described in this paper are the inverse problems of an predictive analysis which is defined by 0‘Connor and Chan [6] as an analysis to simulate the structural response caused by a set of time-varying forces running across a bridge. Two models can be used for this kind of analysis. A. Beam-Elements Model 0‘Connor and Chan [6] model the bridge as an assembly of lumped masses interconnected by massless elastic beam elements as shown in Figure 1, and the nodal responses for displacement or bending moments at any instant are given by Equations (1) and (2) respectively. Moving Loads ... ip. Ip, ^ -D—D—2^ Lumped Masses 1 2 ... N-1 N Figure 1 - Beam-Elements Model {Y} = [Y,]{F}-lY,][Am]{n-m[C]{Y} ( 1 ) {5M} = [5MJ{P}-[5M;][Am]{y}-[5MJ[C]{f} (2) where [P] is a vector of wheel loads, [Am] is a diagonal matrix containing values of lumped mass, [C] is the damping matrix, 5M,L,7,7are the nodal bending moments, displacements, velocities and accelerations respectively, [R,^] {'R ’ can be Y or BAP) is an m x n matrix with the ith column representing the nodal responses caused by a unit load acting at the position of the ith wheel load and [i?J {'R' can be Y or BM) is an n x n matrix with the ith column representing the nodal responses caused by a unit load acting at the position of ith internal node. 1084 B. Continuous Beam Model Assuming the beam is of constant cross-section with constant mass per unit length, having linear, viscous proportional damping and with small deflections, neglecting the effects of shear deformation and rotary inertia (Bernoulli-Euler’s beam), and the force is moving from left to right at a constant speed c, as shown in Figure 2, then the equation of motion can be written as P d-v{xj) a- a + EI ^v(x,0 a- S(x-ct)f{t) (3) where v(x,0 is the beam deflection at point x and time t; p is mass per unit length; C is viscous damping parameter; £ is Young's modulus of material; I is the second moment of inertia of the beam cross-section; f(t) is the time- varying force moving at a constant speed of c, and Sft) is Dirac delta function. \/(0 —O Figure 2 - Simply supported beam subjected to a moving force f(t) Based on modal superposition, the dynamic deflection y(x,t) can be described as follows; v{x,t) = Y,<i'„{x)q„{t) }}=\ (4) where n is the mode number; 0n(x^) is the mode shape function of the n-th mode and qyi(t) is the n-th modal amplitudes. Based on the above mentioned predictive analyses, four Moving Force Identification Methods (MFIM) are developed. 1085 FIRST MOVING FORCE IDENTIFICATION METHOD INTERPRETIVE METHOD I - BEAM-ELEMENTS MODEL (IMI) It is an inverse problem of the predictive analysis using beam-elements model From Equation (1), it can be seen that if Y is known at all times for all interior nodes, then 7 and Y can be obtained by numerical differentiation. Equation (1) becomes an overdetermined set of linear simultaneous equations in which the P may be solved for them. However a particular difficulty arises if measured BM are used as input data. Remembering that the moving loads P are not normally at the nodes, the relation between nodal displacements and nodal bending moments is {r}^[Y,]{BM} + [Yc]{B} ( 5 ) where allows for the deflections due to the additional triangular bending moment diagrams that occur within elements carrying one or more point loads P. [}^]can be calculated from the known locations of the loads. [}^^] and {BM}aic known, but {7} cannot be determined without a knowledge of (P}. 0‘Connor and Chan [6] describe a solution uses values of {P} assumed from the previous time steps. SECOND MOVING FORCE IDENTIFICATION METHOD INTERPRETIVE METHOD II - CONTINUOUS BEAM MODEL (IMII) From the predictive analysis using continuous beam model, if the ith- iTTX mode shape function of the simply supported Euler‘s beam is sin-—-, then the solution of Equation (3) takes the form v = ^sin—F;(0 (6) /=i L where V.(t ), (i = 1, 2, • • •) are the modal displacements. Substitute Equation (6) into Equation (3), and multiply each term of Equation (3) by the mode shape function sin(y;r.x / L ), and then integrate the resultant equation with respect to x between 0 and L and use the boundary conditions and the properties of Dirac function. Consequently, the following equation can be obtained 1086 op u 2 El ^ where = — 5 —- C(y, = C 2/rryy, at the j-th mode. If there are more than one moving loads on the beam, Equation (7) can be written as . 7r(ct-Xk) ._7r(ct~x,) 7r{ct-x^) ■ ■^.1 '2CM' 'cojv; sin-^ L sin-^ • L sin- — L 'K V, 2C,co,K + CO IVj _ 2 . 27r(ct-Xt) sin . 27r(ct~x^} sm-=- ■ . 27r{ct - x .) •• sm—- — Pi ~ Ml A .1 _2C,.a)/„_ _coX_ . n7r(ct-x,) sin—^^ . riKict-x^) sin—^- — . . n7c{ct-x^) •• sin—- — A. L L L ( 8 ) in which is the distance between the k-th load and the first load and x, = 0 . If Pi Pk are constants, the closed form solution of Equation (3) is / N ^ 1 • ■ j7E{ct-Xi) a . . . , ^ {Q\ 48E/f:i' L \ L j ^ J nc in which a = Lcd If we know the displacements of the beam at x,, x,, • • •, , the moving loads on the beam are given by {v} = [S,„.]{/>} (10) in which {v}=[v| v, ■■■ v,]' {•^’} = [-^1 ^2 ■" ^n- , where L" A 1 . j7rx,J . J7r{ct-x.) a . , . , , = 7577X^2772- 27 ^““ 7 ~ -7-/c) 48£:/^;-0 --a ) L \ L j 1087 If l>k, that means the number of nodal displacements is larger than or equal to the number of axle loads, then according to the least squares method, the equivalent static axle load can be given by {/>}=([s,,]''[s„])''[s,q^{v} (11) If the loads are not constant with time, then central difference is used to proceed from modal displacements to modal velocities and accelerations. Equation (8) becomes a set of linear equations in which P^. for any instant can be solved by least squares method. Similar sets of equations could be obtained for using bending moments to identify the moving loads. THIRD MOVING FORCE IDENTIFICATION METHOD TIME DOMAIN METHOD (TDM) This method is based on the system identification theory [7]. Substituting Equation (4) into Equation (3), and multiplying each term by 0j(x), integrating with respect to x between 0 and L, and applying the orthogonality conditions, then dt- dt M., ( 12 ) where con is the modal frequency of the n-th mode; is the damping ratio of the n-th mode; Mn is the modal mass of the n-th modt, pn(t) is the modal force and the mode shape function can be assumed as 0„(x) = sin(^;w / L ). Equation (12) can be solved in the time domain by the convolution integral, and yields = ( 13 ) ^11 0 where ~ sin(^u„0, t >0 (14) and co\, (15) Substituting Equation (13) into Equation (4), the dynamic deflection of the beam at point x and time t can be found as v(x,0 = y—^sin^^ sin6>|,(^ - '^)sin^^^^/(T)dr (16) fxpLo),, L i L 1088 A. Force Identification from Bending Moments The bending moment of the beam at point x and time t is m{xj) = -El- 3c- (17) Substituting Equation (16) into Equation (17), and assuming the force f(t) is a step function in a small time interval and f(t) =0 at the entry and exit, then let ^ 2El7r' n~ . yitu: ^ C,„ = —73 -^sm—Ar, pi co„ L {k) = sin(<:y'„ A/A:), ^ . ,n7(cNi 52(*) = sm(—^/c) (18) (19) N„=- Equation (17) can be expressed as cEt m(2)] 0 ... 0 ' /(i) m(3) < ■=S c,„ /) = 1 E;,S,{2)S,i\) e:S,{\)S,{2) ... 0 /(2) m{N) _E::"S,iN-\)S,il) E;^--S,{N~2)S,i2) Kc. ( 20 ) where A^ is the sample interval and N+I is the number of sample points, and f>„=£r'''“*'S,(Al-A^„+l)S,(//„-l) Equation (20) can be simplified as B f = m ( 21 ) (A'-l)x(;V/^-|) ('V/^->)xl (,V_l)xl U N = matrix B is a lower triangular matrix. We can directly find the force vector f by solving Equation (21). If 77 > and/or 77/ bending moments (77/ > 1) are measured, least squares method can be used to find the force vector f from 1089 ( 22 ) B, ■ m, f m. ®-v,. The above procedure is derived for single force identification. Equation (21) can be modified for two-forces identification using the linear superposition principle as “B. 0 B, B, B. B, where B 3 [N^x (Nb-1)] , Bt [(N -1-2 NJx (Ng-l)], and B^ [NjX (Nb -1)] are sub-matrices of matrix B. The first row of sub-matrices in the first matrix describes the state having the first force on beam after its entry. The second and third rows of sub-matrices describe the states having two-forces on beam and one force on beam after the exit of the first force. B. Identification from Bending Moments and Accelerations Similarly the acceleration response of the beam can be expressed as A f = V (24) The force can also be found from the measured acceleration from Equation (24). If the bending moments and accelerations responses are measured at the same time, both of them can be used together to identify the moving force. The vector m in Equation (21) and v in Equation (24) should be scaled to have dimensionless unit, and the two equations are then combined together to give A/iv||J [v/llvlj (25) where Ihli is the norm of the vector. FOURTH MOVING FORCE IDENTIFICATION METHOD FREQUENCY DOMAIN METHOD (FDM) Equation (12) can also be solved in the Frequency Domain. Performing the Fourier Transform for Equation (12), 1090 1 1 (26) where co;, -CO- +24„co„co M„ W i —00 (27) (28) Let _1_ co;,-co- +2^„a„a (29) Hn(co) is the frequency response function of the n-th mode. Performing the Fourier Transform of Equation (4), and substituting Equations (26) and (29) into the resultant equation, the Fourier Transform of the dynamic deflection v(x,t) is obtained as A. Force Identification from Accelerations Based on Equation (30), the Fourier Transform of the acceleration of the beam at point .x and time t can be written as V(x,a) = -®^X^(D„(x)//„(cy)P„(cy) (31) Considering the periodic property of the Discrete Fourier Transform (DFT), and let A/^m- Equation (32) can be rewritten as (32) 1091 ,m = 0,l, (33) V{m) = X H„(m)'i'„(m)[F,{0) + iF,(0)] /;=1 A^/2-1 « _ k=\ H=t Nn~\ =0 _ /t = l /) = l +E /2)[F„(A^ /2) - iivCA^ / 2)] «=I where is the Fourier Transform of the n-th mode shape, and F is the Fourier Transform of the moving force. Writing Equation (33) into matrix form and dividing F and V into real and imaginary parts, it yields Because F, (0) = 0, Fj (N / 2) = 0,Vj (0) = 0,F; (/ 2) = 0, Equation (34) can be condensed into a set of N order simultaneously equations as (35) Fr and F, can be found from Equation (35) by solving the Nth order linear equation. The time history of the moving force f(t) can then be obtained by performing the inverse Fourier Transformation. If the DFTs are expressed in matrix form, the Fourier Transform of the force vector f can be written as follows if the terms in f are real [8]. F = —Wf (36) N whereW = e''^‘‘^^^ (37) 1092 0 0 0 0 0 0 1 2 ••• -2 -1 0 2 4 ••• -4 -2 0 -2 -4 ••• 4 2 0 -1 -2 2 1 yv X The matrix W is an unitary matrix, which means w-‘ =(w* y (38) where W* is a conjugate of W. Substituting Equation (36) into Equation (35), V = —Al N W 0 -1 X _Ij (39) or V=^ A fe (40) linking the Fourier Transform of acceleration V with the force vector fg of the moving forces in the time domain. Wg is the sub-matrix of W. If N = fg can be found by solving the Mh order linear equations. If N> or more than one accelerations are measured, the least squares method can be used to find the time history of the moving force f(t). Equation (40) can be rewritten as follows (41) relating the accelerations and force vectors in the time domain. Also if N - Nb, fe can be found by solving the Nth order linear equation. If N > Nb or more than one acceleration are measured, the least squares method can be used to find the time history of the moving force f(t). If only Nc {Nc ^ N) response data points of the beam are used, the equations for these data points in Equation (41) are extracted, and described as V, =(w*)" A C V B / A/x/V c B Nr X N W 3 /Vx;V, yV„x! (42) 1093 In usual cases Nc > Ng, so the least squares method is used to find the time history of the moving force f(t). More than one acceleration measurements at different locations can be used together to identify a single moving force for higher accuracy. B. Identification from Bending Moments and Accelerations Similarly, the relationships between bending moment m (and M) and the moving force f can be described as follows, M = ^ B W is (43) /Vxl N N^Nn (44) (45) The force vector fg can be obtained from the above three sets of equations. Furthermore, these equations can be combined with Equations (40), (41) and (42) to construct overdetermined equations before the equations are scaled. Two forces identification are developed using the similar procedure as that for the Time Domain Method. COMPARATIVE STUDY The first moving force identification method is implemented in a computer program using FORTRAN, while the other three methods are implemented under the environment of a high performance numerical computation and visualization software. The predictive analysis using beam- elements model is used to generate the theoretical bridge responses and the four moving force identification methods then use these responses to recover the original dynamic loads. In this study, if at least 80% of the identified forces at any instant of any load lie within ± 10% of the original input force, the method is considered acceptable. It is found that all the four methods can give acceptable results. It is decided to carry out a preliminary comparative study on the four methods in order to study the merits and limitations of each method so as to consider the future development of each method and devise a plan to develop a 1094 moving force identification system which can make use of the benefits of all the four methods. A. Maximum Number of Forces This is to examine the maximum number of axle loads that can be identified by each method. Theoretically, provided that sufficient number of nodal sensors are installed, IMI and IMII can be used to identify as many loads as the system allows. Basically, the number of axle loads cannot be larger than the number of nodal sensors. Regarding TDM and FDM, as the formulation of the governing equation is derived for two moving forces, the maximum number of axle loads that can be identified is two. B. Minimum Number of Sensors Based on a study of common axle spacings of vehicles currently operating on Australian roads, and the cases with zero nodal responses, 0‘Connor and Chan [6] state the relationships of the minimum number of sensors used for IMI and the span length of a bridge as follows: Using bending moment, for span length L > 4.8m, Min. number of nodal moments required = int{ Using displacements, for span length L > 13.8m, Min. number of nodal displacement required = INj{ and for span length L <13.8m, Min. number of nodal displacement required L - 4.8 ' 1.7 L-UX 3.7 + 4 + 6 (46) (47) (48) For IMII, it is found that the number of sensors required are generally less than that for IMI. Regarding TDM and FDM, the programs are not as flexible as that for IMI and IMII and it is not easy to change the number of sensors. Meanwhile the sensors are fixed to be at 1/4, 1/2 and 3/4 of the span. C. Sensitivity towards Noise In general, all the four methods can compute the identified forces exactly the same as those given to the predictive analysis to generate the corresponding responses. It is decided to add white noise to the calculated responses to simulate polluted measurements and to check their sensitivity towards noise. The polluted measurements are generated by the following 1095 equations: m = n>o.nkui«.«] x N„, (49) where Ep is a specified error level; is a standard normal distribution vector (with zero mean value and unity standard deviation). Several cases are studied using Ep =1%, 3%, 5% and 10%. It is found that when using bending moments for IMI and IMII, and if Ep is less than 3%, acceptable results can be obtained. For noise which is greater than 3%, a smoothing scheme should be adopted to smooth the simulated data. Acceptable results cannot be obtained for Ep > 10%. Besides, both IMI and IMII cannot give acceptable results when using displacements. Both TDM and FDM cannot give acceptable results when using displacements only, accelerations only or bending moments only. In general TDM and FDM are less sensitive to noise when comparing to IMI and IMII. They can give acceptable results for Ep up to 5 % without any smoothing of the polluted simulated data. D. Computation Time In general, the computer program for IMI only takes few seconds to identify moving forces. In order to compare the computation time, IMI is implemented in the same environment as the other three methods. It is found that IMI and IMII take about 2-3 minutes to give the identified forces for a case of two axle loads using a 80486 computer. However, under the same working conditions, TDM and FDM almost take a whole day for any one of them to identify two moving forces. It is due to the fact that both of them require to set up an huge parametric matrix. CONCLUSIONS Four methods are developed to identify moving time-varying force and they all can produce acceptable results. From a preliminary comparative study of the methods, it is found that IMI and IMII have a wider applicability as the locations of sensors are not fixed and it can identify more than two moving forces. However, TDM and FDM are less sensitive to noise and require less number of sensors. It is decided to further improve the four methods and then a more detailed and systematic comparison can be carried out afterwards. The possible development of the methods are described as follows. 1096 Both the IMI and IMII are developed to work with one kind of responses, e.g. either displacements or bending moments. It is suggested to modify the programs to use mixed input parameter, e.g. use bending moments as well as accelerations as that for TDM or FDM. Regarding the TDM and FDM, as the basic formulations are based on two-axle moving forces, so it is necessary to modify the governing equations for multi-axle. In addition, the computation time for TDM or FDM under the environment of the high performance numerical computation and visualization software used is unbearable. It is expected that the time will be significantly reduced if the methods are implemented in programs using standard programming languages like FORTRAN 90 or C. Then the four methods can be combined together and merged into a Moving Force Identification System (MFIS) so that it can automatically select the best solution routines for the identification. ACKNOWLEDGMENT The present project is funded by the Hong Kong Research Grants Council. REFERENCES 1. Davis, P. and Sommerville, F., Low-Cost Axle Load Determination, Proceedings, 13th ARRB & 5th REAAA Combined Conference, 1986, Part 6, p 142-149. 2. Peters, R.J., AXWAY - a System to Obtain Vehicle Axle Weights, Proceedings, 12th ARRB Conference, 1984, 12 (2), p 10-18. 3. Peters, R.J., CULWAY - an Unmanned and Undetectable Highway Speed Vehicle Weighing System, Proceedings, 13th ARRB & 5th REAAA Combined Conference, 1986, Part 6, p 70-83. 4. Cebon, D. Assessment of the Dynamic Wheel Forces Generated by Heavy Vehicle Road Vehicles. Symposium on Heavy Vehicle Suspension Characteristics, ARRB, 1987. 5. Cantineni, R., Dynamic Behaviour of Highway Bridges Under The Passage of Heavy Vehicles. Swiss Federal Laboratories for Materials Testing and Research (EMPA) Report No. 220, 1992, 240p. 6. O'Connor, C. and Chan, T.H.T., Dynamic Wheel Loads from Bridge Strains. Structural Engineering ASCE, 114 (STS), 1988, p. 1703-1723. 7. Briggs, J.C. and Tse, M.K. Impact Force Identification using Extracted Modal Parameters and Pattern Matching, International Journal of Impact Engineering, 1992, Vol. 12, p361-372. 8. Bendat, J.S. and Piersol, J.S., Engineering Application of Correlation and Spectral Analysis. John Wiley & Sons, Inc. Second Edition, 1993. 1097 1098 ESTIMATING THE BEHAVIOUR OF A NONLINEAR EXPERIMENTAL MULTI DEGREE OF FREEDOM SYSTEM USING A FORCE APPROPRIATION APPROACH P.A. Atkins J.R.Wright Dynamics and Control Research Group School of Engineering, Simon Building, University of Manchester, Oxford Road, Manchester. M13 9PL ABSTRACT The identification of nonlinear multi degree of freedom systems involves a significant number of nonlinear cross coupling terms, whether the identifi¬ cation is carried out in spatial or modal domains. One possible approach to reducing the order of each identification required is to use a suitable pattern of forces to drive any mode of interest. For a linear system, the force pattern required to drive a single mode is derived using a Force Ap¬ propriation method. This paper presents a method for determining the force pattern necessary to drive a mode of interest of a nonlinear system into the nonlinear region whilst the response is controlled to remain in pro¬ portion to the linear mode shape. Such an approach then allows the direct nonlinear modal terms for that mode to be identified using the Restoring Force method. The method for determining the relevant force patterns is discussed. The implementation of the method for experimental systems is considered and experimental results from a two degree of freedom ’bench¬ mark structure’ are presented. INTRODUCTION Force Appropriation [1] is used in the analysis of linear systems to de¬ termine the force patterns which will induce single mode behaviour when applied at the relevant natural frequency. This technique is used in the aerospace industry during Ground Vibration Tests: each normal mode of 1099 a structure is excited using the derived force pattern and thus identified in isolation. Current practice, when the presence of nonlinearity is sus¬ pected, is to increase input force levels and monitor the variation of tuned frequencies. Some information about the type of nonlinearity present may be found, but no analytical model can be derived. Thus predictions for behaviour at higher levels of excitation axe not possible. A number of techniques for identifying nonlinearity, for example the Restor¬ ing Force method [2], have been demonstrated on systems with low num¬ bers of degrees of freedom. Unfortunately, in practice, structures have a large number of degrees of freedom, often with a high modal density. A classical Restoring Force approach to the identification of such systems could involve a prohibitive number of cross coupling terms. The ability to treat each mode separately, by eliminating the effects of the cross coupling terms, would thus.be advantageous. Subsequent tests could then evaluate the cross coupling terms. For these reasons it would be useful to extend Force Appropriation to the identification of nonlinear systems. An approach has been developed [6] that allows an input force pattern to be derived that will result in a non¬ linear response in the linear mode shape of interest. This force pattern is derived using an optimisation approach. The mode of interest can then be identified using a single degree of freedom nonlinear identification method. In this work the Restoring Force method is used to examine the nonlinear response of a particular linear mode and an application of this approach to a two degree of freedom experimental system is presented. THEORY The theoretical approach is demonstrated for the two degree of freedom system with spring grounded nonlinearity shown in figure 1. The equations of motion for this system in physical space are: m 0 0 m + + (1 + a)c —ac '±1 1 —ac (1 + d)c \ X2 J (l -}- Q^k —ak fill —ak (1 + a)k l^x\ 0 where is the cubic stifEhess coefficient and a is a constant that allows the frequency spacing of the natural frequencies to be varied. These equations can be transformed to linear modal space using the transformation: {i} = [ij>]{u} (2) where [(j>] is the modal matrix of the underlying linear system and the vector {u} defines the modal displacements. For this symmetrical system 1100 the modal matrix is 1 1 1 -1 (3) The equations of motion transformed to linear modal space using the nor¬ malised modal matrix are: 771 0 \ ill 0 771 (112 + c 0 0 (1 + 2g)c 1 ^ 2 / ■jb 0 Ull 0 (1 + 2a)k 1 ^ 2 ] P{Ul-U2y/4:' -U2)V4 where {p} is the modal input vector. It can be seen from the above equa¬ tions that the cubic nonlinearity couples the modes in linear mod^ space; in fact there are a significant number of terms for a single nonlinearity. The proposed method aims to determine the force pattern that will reduce the response of this system to that of a single mode. It was shown in a previous paper [3] that this can be achieved by seeking a force vector that will cause motion only in the target mode, by eliminating motion in the coupled mode. In practice, physical data from transducers are available. Any subsequent transformations would be time consuming. It is shown below that causing motion in one mode to be zero is equiva¬ lent to forcing motion in a linear mode shape, mode one in this example. Consider the coordinate transformation {a;} = [<^]{ii} or more explicitly for the two degree of freedom system in Figure 1: Thus 1 2 ‘l -1 1 1 (5) ( 6 ) and enforcing the first mode shape (1,1) in physical space should give a second modal displacement of zero. 1 2 1 -1 1 1 1 1 (7) So if an excitation is applied which causes the nonlinear system to vibrate in its first linear mode shape, the response will be composed only of ui and the influence of the coupled mode, U 2 , will then have been eliminated. The method must therefore derive a force pattern which will cause the system to vibrate in one of its linear mode shapes. It has been shown 1101 in a previous paper [3] that if the response contains harmonics then the force pattern must also contain harmonics in order to control the harmonic content of the response. In theory, the responses will be an infinite series of harmonics, but this series is truncated in this case of a cubic stiffness nonlinearity to include only the fundamental and third harmonic terms. The physical input forces will thus be of the form: fi(t) = Fii cos(a;ea;t 4- ^ii) 4- F 13 cos(a;ea;t + (^ 13 ) ( 8 ) f2{t) = F21 C0s(a;ea;t + (j>2l) 4* F23 COS{uJext + fe) (9) where is the excitation frequency. Parameters for these force patterns may then be chosen such that only mode one is excited. OPTIMISATION APPROACH In general, no a priori model of the system exists so an optimisation routine is used to determine the force pattern parameters required to maximise the contribution of the mode of interest. The objective function, the quantity that the optimisation routine seeks to minimise, must be representative of the deviation of the response from the target linear mode shape. The objective function, F, that was chosen in this case was based on the vector norm [4] of the two physical responses, Xi and X 2 , and is shown below: where and 02 are elements of the mode shape vector for the target mode. This summation is carried out over one cycle of the fundamental response. The number of data points per cycle is npts and Xki the kth response at the itk sample. This objective function allows the response to contain harmonics and can be extended to more degrees of freedom by choosing a reference displacement and subtracting further displacements from it. The Variable Metric optimisation method [5] was used in this work as it has been found to produce the best results for simulated data. The application of this method to a two degree of freedom system such as that shown in figure 1 is detailed in [6]. Optimised force patterns are obtained at several levels of input amplitude. These force patterns are then applied and the Restoring Force method is used to curve fit the resulting modal displacement and velocity time histories to give the direct linear and nonlinear coefficients for the mode of interest. 1102 EXPERIMENTAL IMPLEMENTATION The simulated application of this method assumed that certain parame¬ ters were known. In order to carry out an identification of an experimental structure, these parameters must be measured or calculated. Some pro¬ cessing of experimental data is necessary in order to apply the Restoring Force method. The restoring force of a system can be expressed for a single degree of freedom system as: h{x, x) = f(t) - mx (11) where h{x, x) is the restoring force and f{t) the input force. A similar ex¬ pression applies to the modal restoring force for an isolated mode. Thus the input force, acceleration, velocity and displacement must be calculated at the each time instant. A similar expression applies to the modal restoring force for an isolated mode. In the experimentaJ case it is usual to measure acceleration and input force; the remaining two states must therefore be obtained by integration of the acceleration time history. Frequency domain integration [7] was used for this purpose. High pass filtering was used to remove any low frequency noise which can be amplified by this type of integration. Several methods have been suggested for estimating modal mass, but in this study a method developed by Worden and Tomlinson [8] was used. An estimate for the modal mass is obtained and then an error term is included in the curve fit which will iteratively yield a more accu¬ rate estimate of the mass. Generally the mass value will converge after one iteration. The objective function used in the simulations was calculated from the displacement time histories. In the experimental case, acceleration was used rather than displacement as it was considered that using ’raw’ data would be quicker and give less opportunity for error. In the simulated case, the system parameters were known a priori so the modal matrix of the underlying linear system could be calculated. For most types of nonlinearity the response of the system at low input force levels will be dominated by linear terms. Normal mode tuning [1] was therefore applied at low force levels to yield an approximation to the modal matrix of the underlying linear system. A quality indicator to give some idea of the effectiveness of the optimisa¬ tion performed would be advantageous. Results corrupted by background noise, for example, could then be discarded. A perfect optimisation will occur when the ratio of measured accelerations exactly matches the mode shape ratio specified for the mode of interest. Thus a least squares fit of the sampled accelerations was carried out over a cycle of the fundamental frequency and the percentage error of the measured mode shape to the 1103 required mode shape was calculated. This percentage error will indicate whether the optimisation has been successful. To assess the accuracy of the parameters estimated using this method, an identification was carried out using a conventional Restoring Force ap¬ proach in physical space. A band limited random excitation was used, and the physical data processed and curve fitted. The physical parameters were then transformed to modal space. The direct linear and nonlinear param¬ eters for modes one and two are shown in table 1. It should be noted that although this conventional Restoring Force approach is possible for this two degree of freedom system, it will not generally be possible since the number of terms in the curve fit increases dramatically when different t 3 q)es of nonlinearity and more degrees of freedom are included. It is carried out in this case as a means of validating the proposed method. EXPERIMENTAL SETUP The rig constructed consisted of two masses on thin legs connected in series by a linear spring, each mass being driven by a shaker. A cubic nonlin¬ earity was introduced between the first mass and ground using a clamped- clamped beam attached at the centre which will yield a cubic stiffness for large deflections [9], A schematic diagram of the rig is shown in figure 2. The force input by each shaker was measured using a force gauge and the acceleration of each mass was measured using an accelerometer in the po¬ sitions also shown in figure 2, Acceleration and force data were acquired using a multiple channel acquisition system, the optimisation routine was carried out on line. RESULTS Normal mode tuning of the rig gave natural frequencies of 20.67 Hz and 24.27 Hz and a modal matrix of: 3.87 5.03 5.52 -3.27 ( 12 ) The excitation frequency was chosen to be slightly lower than the natural frequency of the mode of interest in order to avoid the problems associated with force drop out which are worst at the natural frequency. For each mode optimisation was performed at three input force levels, the highest level was as high as possible so as to excite the nonlinearity strongly. The optimisation routine was carried out using the voltage input into the signal generator as the variable. The force input into the structure was measured 1104 for use in the Restoring Force identification but was not used in the opti¬ misation as it is not directly controllable. The details of the optimisation for each force level are presented in table 2. The optimised forces and re¬ sulting accelerations for mode one are shown in figure 3 and figure 4. The acceleration data for the optimised force patterns were then integrated and the modal restoring force for the mode of interest calculated. The initial estimate of modal mass for the calculation of the modal restoring force was tahen from a previous paper [10] in which the rig was identified using a using a physical parameter identification method. The mass was estimated in this paper to be 2.62 kg, this physical mass will then be equal to the modal mass since the modal matrix was normalised to be orthonormal. The restoring force data was then transformed to modal space. The modal restoring force surface obtained using optimised force inputs for mode one is shown in figure 5 and a stiffness section through this surface is shown in figure 6. The restoring force time histories were then curve fitted against modal velocity and displacement. The estimated parameters for mode one axe shown in table 3. It can be seen that they do not compare very well with those estimates obtained using the band limited random excitation. It was suspected that at lower excitation levels the estimates were being distorted by linear dependence [11]. Linear dependence is a problem which occurs when curve fitting a harmonic response from a linear system; the equations of motion may be identically satisfied by mass and stiffness terms modified by an arbitrary constant. This condition is avoided by the harmonic terms introduced into the response by nonlinearity. The curve fit was thus repeated using only the data obtained from the highest level of excitation; the estimates obtained are shown in table 4. It can be seen that the linear parameter estimates now agree well with the band limited random results. The estimates for the cubic stiffness coefficient do not appear to agree so well. The standard deviation on the cubic stiffness derived from the band limited random excitation is approximately a third of the value of the parameter itself. The uncertainty on this parameter occurs because the nonlinearity is not very strongly excited by this type of excitation. A stiffness section through the restoring force surface, figure 7, shows little evidence of a cubic stiffness component. If a higher level of excitation were possible then a better estimate may be achieved. The identification w^ repeated for mode two. The restoring force surface obtained and a stiffness section through it are shown in figure 8 and figure 9. It can be seen from the stiffness section that the nonlinearity is not very strongly excited. The estimated direct modal parameters are shown in table 5. It can be seen that these results agree quite well with those obtained using band limited random excitation. It is considered that the discrepancy between the two sets of results, in particular the mass and 1105 stiifiiess estimates, is again due to linear dependence. CONCLUSIONS An extension of the force appropriation method has been proposed for nonlinear systems. In this method, an optimisation routine is used to determine the force patterns which will excite a single mode nonlinear response . The direct linear and nonlinear modal parameters can then be estimated from a curve fit of the modal restoring force surface. The method was applied to an experimental two degree of freedom system whose modes were coupled in linear modal space by a spring grounded nonlinearity. A conventional restoring force identification was performed using a band limited random signal for comparison. The parameters estimated from the single mode responses were found to agree quite well with those from the band limited random tests. REFERENCES 1. Holmes P., Advanced Applications of Normal Mode Testing, PhD Thesis, University of Manchester 1996. 2. Hadid M.A. and Wright J.R., Application of Force State Mapping to the Identification of Nonlinear Systems Mechanical Systems and Signal Processing, 1990, 4(6), 463-482 3. Atkins P.A., Wright J.R., Worden K., Manson G.M. and Tomlinson G.R., Dimensional Reduction for Multi Degree of Freedom Nonlinear Systems, International Conference on Identification in Engineering Systems 1996, 712-721 4. Kreyszig E., Advanced Engineering Mathematics, Wiley, 6th Edition 5. Press W.H., Teukolsky S.A., Vettering W.T. and Flannery B.P., Nu¬ merical Recipes in Fortran Cambridge University Press, 6th Edition 6. Atkins P.A. and Wright J.R., An Extension of Force Appropriation for Nonlinear Systems Noise and Vibration Engineering, Proceedings of ISMA21(2), 915-926, 1996 7. Worden K., Data Processing and Experiment Design for the Restor¬ ing Force Method, Part I: Integration and Differentiation of Measured Time Data, Mechanical Systems and Signal Processing, 4(4) 295-319, 1990 1106 8. Ajjan Al-Hadid M., Identification of Nonlinear Dynamic Systems us¬ ing the Force State Mapping Technique, PhD Thesis, Queen Mary College, University of London, 1989 9. Storer D.M., Dynamic Analysis of Nonlinear Structures Using Higher Order Frequency Response Function, PhD Thesis, University of Manch¬ ester, 1991 10. Atkins P. and Worden K., Identification of a Multi Degree of Freedom Nonlinear System, Proc. of IMAC XV, 1997 ACKNOWLEDGEMENTS This work was supported by E.P.S.R.C. under research grant number GR/J48238 at the University of Manchester Modal parameter Mode one Mode two k (N/m) c (Nm/s) /3(N/m^) m (kg) 4.87 X 10^ 10.11 3.83 X 10® 2.60 6.49 X 10^ 9.49 8.90 X 10* 3.06 Table 1: Direct modal parameters estimated from curve fit of band limited random data Low forcing Medium forcing High forcing Fii (Volts) 1.0 2.0 3.0 F 21 (Volts) 1.0 2.0 3.0 u)^ (Hz) 20.0 20.0 20.0 initial mode shape ratio 0.72 0.68 0.65 final mode shape ratio 0.70 0.70 0.70 target mode shape ratio 0.70 0.70 0.70 percentage error 0.14 0.03 0.71 Table ,2: Details of optimisation for mode one Table 3: Direct modal parameters estimated from optimised responses Model parameter Estimated parameter k (N/m) 4.57 X 10^ c (Nm/s) 8.62 /?(N/m3) 6.81 X 10® m (kg) __2^75_ Table 4: Direct modal parameters estimated using high force level only Model parameter Estimated parameter k (N/m) c (Nm/s) m (kg) 5.37 X 10^ 10.16 2.45 X 10® 2.29 Table 5: Direct modal parameters estimated for mode two Figure 1: Two degree of freedom system 1108 0 s 0.40008 Figure 3: Optimised forces for mode one at a high force level Figure 4: Accelerations responses to optimised forces Figure 5: Modal restoring force surface for mode one Force (N) Figure 7: Stiffness section through modal restoring force derived from ran¬ dom excitation for mode one 1111 Figure 8: Modal restoring force surface for mode two Figure 9: Stiffness section through modal restoring force for mode two 1112 POWER FLOW TECHNIQUES II THE OPTIMAL DESIGN OF NEAR-PERIODIC STRUCTURES TO MINIMISE NOISE AND VIBRATION TRANSMISSION R.S. Langley, N.S. Bardell, and P.M. Loasby Department of Aeronautics and Astronautics University of Southampton Southampton SO 17 IBJ, UK 1. INTRODUCTION An engineering structure is said to be of "periodic" construction if a basic structural unit is repeated in a regular pattern . A beam which rests on regularly spaced supports is one example of a one-dimensional periodic structure, while an orthogonally stiffened cylinder is an example of a two- dimensional periodic structure. It has long been known that perfectly periodic structures have very distinctive vibration properties, in the sense that "pass bands" and "stop bands" arise: these are frequency bands over which elastic wave motion respectively can and cannot propagate through the structure [1,2]. If the excitation frequency lies within a stop band then the structural response tends to be localised to the immediate vicinity of the excitation source. Conversely, if the excitation frequency lies within a pass band then strong vibration transmission can occur, and it is generally the case that the resonant frequencies of the structure lie within the pass bands. Much recent work has been performed concerning the effect of random disorder on a nominally periodic structure (see for example [3-5]). It has been found that disorder can lead to localisation of the response even for excitation which lies within a pass band, and this reduces the propensity of the structure to transmit vibration. This raises the possibility of designing disorder into a structure in order to reduce vibration transmission, and this possibility was briefly investigated in reference [6] for a one-dimensional periodic waveguide which was embedded in an otherwise infinite homogeneous system. The present work extends the work reported in reference [6] to the case of a finite near-periodic beam system, which more closely resembles the type of optimisation problem which is. likely to occur in engineering practice. The beam is taken to have N bays, and the design parameters are taken to be the individual bay lengths. Both single frequency and band-limited excitation are considered, and two objective functions are investigated: (i) the response in a bay which is distant from the applied loading (minimisation of vibration transmission), and (ii) the maximum response in the structure (minimisation of maximum stress levels). In each case the optimal configuration is found by employing a quasi-Newton algorithm, and the physical features of the resulting design are discussed in order to suggest general design guidelines. 1113 2. ANALYTICAL MODEL OF THE NEAR-PERIODIC BEAM 2.1 Calculation of the Forced Response A schematic of an N bay near-periodic beam structure is shown in Figure 1. The structure is subjected to dynamic loading, and the aim of the present work is to find the optimal design which will minimise a prescribed measure of the vibration response. No matter what type of optimisation algorithm is employed, this type of study requires repeated computation of the system dynamic response as the design parameters are varied, and it is therefore important to employ an efficient analysis procedure. In the present work the h-p version of the finite element method (FEM) is employed: with this approach the structure is modelled as an assembly of elements which have both nodal and internal degrees of freedom. Each element has two nodes and the nodal degrees of freedom consist of the beam displacement and slope; the internal degrees of freedom are generalised coordinates which are associated with a hierarchy of shape functions which contribute only to the internal displacement field of the element. The internal shape functions used here are the K-orthogonal Legendre polynomials of order four onwards - full details of the present modelling approach are given in reference [7]. For harmonic excitation of frequency co the equations of motion of the complete beam structure can be written in the form where M and K are the global mass and stiffness matrices (assembled from the individual element matrices taking into account the presence of any mass or spring elements and allowing for constraints), q contains the system generalized coordinates, F is the generalized force vector, and r| is the loss factor, which in the present study is taken to be uniform throughout the structure. Equation (1) can readily be solved to yield the system response q. In the present work it is convenient to use the time averaged kinetic and strain energies of each of the N bays as a measure of the response - for the nth bay these quantities can be written as and say, where r,=(£0V4)9;>„?„. (2.3) Here and K„ are the mass and stiffness matrices of the nth bay, and q„ is the vector of generalized coordinates for this bay. Many of the physical features of the forced response of a near-periodic structure can be explained in terms of the free vibration behaviour of the 1114 associated perfectly periodic structure. The following section outlines how the present finite element modelling approach can be used to study the pass bands and stop bands exhibited by a perfect periodic structure. 2.2 Periodic Structure Analysis The finite element method described in section 2.1 can be applied to a single bay of a perfectly periodic structure to yield an equation of motion in the form Dq=F, D=-coW--(l+/r|)ii:, (^’5) where the matrix D is referred to as the dynamic stiffness matrix. In order to study wave motion through the periodic system it is convenient to partition D, q and F as follows r Du ft: D= D„ D,r . F= 0 Dr, F„ V V ( 6 - 8 ) where L relates to the coordinates at the left most node, R relates to those at the right most node, and / relates to the remaining "internal" coordinates. Equations (4-8) can be used to derive the following transfer matrix relation between the displacements and forces at the left and right hand nodes V / V / (9,10) Equation (9) can now be used to analyze wave motion through the periodic system: such motion is governed by Bloch’s Theorem, which states that i^L Fi)=exp{-ie-b){qii -F^) where 8 and 5 are known respectively as the phase and attenuation constants. A pass band is defined as a frequency band over which 6=0, so that wave motion can propagate down the structure without attenuation. It follows from equation (9) that (T-Ie -ie-6' ( \ 0 F, io V V y ( 11 ) SO that 8 and 6 can be computed from the eigenvalues of T, thus enabling the pass bands and stop bands to be identified. 1115 2.3 Optimisation Procedure Equations (l)-(3) enable the forced response of the system to be calculated for any prescribed set of system properties. The aim of the present analysis is to compute the optimal set of system properties for a prescribed design objective, and in order to achieve this equations (l)-(3) are evaluated repeatedly as part of an optimisation algorithm. As an example, it might be required to minimise the kinetic energy of bay N by changing the various bay lengths. In this case equations (l)-(3) provide the route via which the objective function (the kinetic energy in bay N) is related to the design parameters (the bay lengths), and the optimisation algorithm must adjust the design parameters so as to minimise the objective function. The optimisation process has been performed here by using the NAg library routine E04JAF [8], which employs a quasi-Newton algorithm. This type of algorithm locates a minimum in the objective function, although there is no indication whether this minimum is the global minimum or a less optimal local minimum. The probability of locating the global minimum can be increased significantly by repeated application of the NAg routine using random starts, i.e. random initial values of the design parameters. Numerical investigations have led to the use of 30 random starts in the present work. 3. NUMERICAL RESULTS 3.1 The System Considered The foregoing analysis has been applied to a beam of flexural rigidity El, mass per unit length m, and loss factor r|=0.015, which rests on A+1 simple supports, thus giving an A’-bay near-periodic system. The design parameters are taken to be the bay lengths (i.e. the separation of the simple supports), and the design is constrained so that the length of any bay lies within the range 0.9L^,<1.1L, where L, is a reference length. A non-dimensional frequency Q is introduced such that 0.=(oL,N{m/Er), and the non-dimensional kinetic and strain energies of a bay are defined as T,'-T,^(EI/L^^\F\^) and U,'~U„iEI/L,^\F\'^) where F is the applied point load. As discussed in the following subsections, two objective functions are considered corresponding to minimum vibration transmission and minimum overall response. In all cases the excitation consists of a point load applied to the first bay and the response is averaged over 11 equally spaced point load locations within the bay. For reference, the propagation constants for a periodic system in which all the bay lengths are equal to L, are shown in Figure 2 - the present study is focused on excitation frequencies which lie in the range 23<f2<61, which covers the second stop band and the second pass band of the periodic system. 3.2 Design for Minimum Vibration Transmission In this case the objective function is taken to be the kinetic energy in bay N, so that the aim is to minimise the vibration transmitted along the structure. Three types of loading are considered; (i) single frequency loading with Q=50, 1116 which lies within the second pass band of the ordered structure; (ii) band- limited loading with 40<Q<60, which covers the whole of the second pass band; (iii) band-limited loading with 23<Q<61, which covers the whole of the second stop band and the second pass band. Results for the optimal design under single frequency loading are shown in Table 1; in all cases it was found that the bay lengths were placed against either the upper bound (U=1.1L,) or the lower bound (L=0.9L,), and significant reductions in the energy level of bay N were achieved. In this regard it should be noted that the dB reduction quoted on Table 1 is defined as -101og(r;v/^;^r) where is the kinetic energy in the final bay of the ordered system. The optimal designs shown in Table 1 all tend to consist of a bi-periodic structure in which the basic unit consists of two bays in the configuration LU. The pass bands and stop bands for this configuration are shown in Figure 3, and further, for the optimal 12 bay system is shown in Figure 4 over the frequency range 0<Q<250. By comparing Figures 3 and 4 it is clear why the selected design is optimal - the new bi-periodic system has a stop band centred on the specified excitation frequency Q=50. It can be seen from Figure 4 that the improvement in the response at the specified frequency 0=50 is accompanied by a worsening of the response at some other frequencies. Results for the optimal design under band-limited excitation over the range 40<Q<60 are shown in Table 2. In some cases two results are shown for the optimised "Final Energy": in such cases the first result has been obtained by forcing each bay length onto either the upper (U) or lower (L) bound, while the second result has been obtained by using the NAg optimisation routine. If only one result is shown then the two methods yield the same optimal design. The "bound" result is easily obtained by computing the response under each possible combination of U and L bay lengths - this requires 2^ response calculations, which normally takes much less CPU time than the NAg optimisation routine. It is clear from Table 2 that the additional improvement in the response yielded by the full optimisation routine is minimal for this case. The response curve for the 12-bay system is shown in Figure 5, where it is clear that a significantly reduced response is achieved over the specified frequency range; as would be expected an increase in the response can occur at other frequencies. It is interesting to note that most of the optimal designs shown in Table 2 lack symmetry - however, it follows from the principle of reciprocity that a design which minimises vibration transmission from left to right will also minimise transmission from right to left. It should therefore be possible to "reverse" the designs without changing the transmitted vibration levels. This hypothesis is tested in Figure 6 for a 12 bay structure - the figure shows the energy distribution for the optimal design UUULUULLLLLU and for the reversed design ULLLLLUULUUU. Although the detailed distribution of energy varies between the two designs, the energy levels achieved in bay 1117 12 are identical, as expected. Results for the optimal design under wide-band excitation 23<Q<61 are shown in Table 3, and the response curve for the 12-bay optimised system is shown in Figure 7. The form of optimal design achieved is similar to that obtained for the narrower excitation band 40<Q<60, although there are detailed differences between the two sets of results. In each case there is a tendency for a group of lower bound bays (L) to occur in the mid region of the structure, and a group of upper bound bays (U) to occur at either end. This creates an "impedance mismatch" between the two sets of bays, which promotes wave reflection and thus reduces vibration transmission along the structure. By comparing Tables 1-3, it is clear that the achievable reduction in vibration transmission reduces as the bandwidth of the excitation is increased. 3.3 Design for Minimum ''Maximum” Strain Energy In this case the strain energy U„ of each bay is computed and the objective function is taken to be the maximum value of U„. As a design objective, this procedure can be likened to minimising the maximum stress in the structure. As in the previous section the three frequency ranges Q=50, 40<n<60, and 23<Q<61 are considered, and the present study is limited to systems having 9,10, 11, or 12 bays; the optimal designs achieved are shown in Table 4. Considering the single frequency results (Q.=50) shown in Table 4, it is clear that a large dB reduction is achieved only for those systems which have an even number of bays; furthermore, the optimal energy obtained has the same value (0.0297) in all cases. This can be explained by noting that for an odd number of bays the frequency Q=50 lies near to an anti-resonance of the ordered structure, whereas a resonance is excited for an even number of bays - this feature is illustrated in Figure 8 for the 12 bay structure. The repeated occurrence of the optimal energy 0.0297 arises from the fact that the initial bay pattern ULLLUUU occurs in all four designs - it has been found that this pattern causes a vibration reduction of over 20dB from bay 1 to bay 8, so that the response in bay 1 (the maximum response) is insensitive to the nature of structure from bay 8 onwards. The optimal "bounded" designs arising for band-limited excitation either tend to be of the "UL" bi-periodic type or else nearly all the bays are assigned the same length. However it should be noted that in all cases the design produced by the NAg optimisation routine offers an improvement over the "bounded" design, particularly for the wide-band case (23<n<61). It is clear from Table 4 that the achieved reduction in strain energy reduces as the bandwidth of the excitation is increased. 1118 4. CONCLUSIONS The present work has considered the optimal design of a near-periodic beam system to minimise vibration transmission and also maximum stress levels. With regard to vibration transmission it has been found that very significant reductions in transmission are achievable with relatively minor design changes. The optimum design normally involves placing the design parameters (the bay lengths) on the permissible bounds, and this means that a simple design search routine can be used in preference to a full optimisation algorithm. With regard to minimum stress levels, it has been found that the optimal design for wide¬ band excitation is not normally a "bounded" design, and thus use of a full optimisation algorithm is preferable for this case. For both vibration transmission and maximum stress levels, the benefits obtained from an optimal design decrease with increasing excitation bandwidth, but nonetheless very significant reductions can be obtained for wide-band excitation. REFERENCES 1. S.S. MESTER and H. BENAROYA 1995 Shock and Vibration 2, 69- 95. Periodic and near-periodic structures. 2. D.J. MEAD 1996 Journal of Sound and Vibration 190, 495-524. Wave propagation in continuous periodic structures: research contributions from Southampton 1964-1995. 3. C.H. HODGES 1982 Journal of Sound and Vibration 82, 411-424. Confinement of vibration by structural irregularity. 4. D. BOUZIT and C. PIERRE 1992 Journal of Vibration and Acoustics 114, 521-530. Vibration confinement phenomena in disordered, mono- coupled, multi-span beams. 5. R.S. LANGLEY 1996 Journal of Sound and Vibration 189, 421-441. The statistics of wave transmission through disordered periodic waveguides. 6. R.S. LANGLEY 1995 Journal of Sound and Vibration 188, 717-743. Wave transmission through one-dimensional near periodic structures: optimum and random disorder. 7. N.S. BARDELL, R.S. LANGLEY, J.M. DUNSDON and T. KLEIN 1996 Journal of Sound and Vibration 197, 427-446. The effect of period asymmetry on wave propagation in periodic beams. 8. ANON 1986 The NAg Fortran Workshop Library Handbook - Release 1. Oxford: NAg Ltd. 1119 TABLE 1 Optimal design of 1-D beam structure, to minimise energy transmission, D.=50. Original Energy; Non-dimensional kinetic energy in bay N of the periodic structure. Final Energy; Non-dimensional kinetic energy in bay N of the optimised structure. No. of Optimal Pattern Original Final Reduction Bays, N Energy Energy (dB) 4 UULU 0.276E 1 0.804E-3 35.348 ULULU 0.609E-1 0.179E-3 ■S UULULULU 0.674E 0 0.613E-5 ULULULULU 0.564E-1 0.135E-5 46.216 1— 10 UULULULULU 0.424E 0 0.532E-6 ^ 11 ulululululu 0.535E-1 0.n7E-6 56.604 12 UULULULULULU 0.289E 0 0.461E-7 67.966 13 ULULULULULULU 0.502E-1 O.lOlE-7 66.950 16 UULULULULULULULU 0.154E0 0.346E-9 __— 86.484 17 ululululululululu 0.43 lE-1 0.761E-10 87.529 1 1120 TABLE 2 Optimal design of 1-D beam structure, to minimise energy transmission, 40<Q<60. Original Energy: Non-dimensional kinetic energy in bay N of the periodic structure. Final Energy: Non-dimensional kinetic energy in bay N of the optimised structure. No. of Optimal pattern Original | Final Reduction Bays, N Energy Energy (dB) ■■ ULLU 0.670E 0 0.103E-1 18.112 5 ULLLU 0.63 IE 0 0.735E-2 19.338 0.711E-2 19.482 6 UULLLU 0.221E-2 22.407 7 ULLUULU 0.463E 0 0.171E-2 24.335 8 UULLLLLU 0.430E 0 0.966E-3 26.487 0.914E-3 26.725 9 UUULLLLLU 0.444E 0 0.341E-3 31.142 10 UUUULLLLLU 0.449E 0 0.192E-3 33.681 0.189E-3 33.758 11 ULLUUUULLLU 0.291E0 0.821E-4 35.504 12 UUULUULLLLLU 0.201E0 0.352E-4 37.558 13 1 ULUUUUULLLLLU 0.199E0 0.153E-4 41.148 1121 TABLE 3 Optimal design of 1-D beam structure, to minimise energy transmission, 23<Q<6L Original Energy; Non-dimensional kinetic energy in bay N of the periodic structure. Final Energy: Non-dimensional kinetic energy in bay N of the optimised structure. 6 LLLLUU 7 LLLULUU 8 LLLLLUUU 9 ULLLLLLUU 0.494E 0 0.183E0 0.175E0 0.648E-2 0.246E-2 0.180E-2 III » UUULLLLLUU 0.105E0 0.277E-3 2.762 3.916 18.821 18.715 19.878 21.868 25.787 11 UUULLLLLLUU 0.105E0 0.776E-4 31.313 12 UlTULLLLLLLUU 0.166E0 0.526E-4 34.991 13 UUULLLLLLULUU 0.973E-1 0.282E-4 35.379 14 UUUULLLLLLULUU 0.581E-1 0.122E-4 36.778 TABLE4 Optimal design ofl-D beam structure, to minimise "maximum” strain energy. Bay No.: Bay in which the optimal minimum “maximum” non-dimensional strain energy occurs Original Energy: Initial “maximum” non-dimensional bay strain energy of the periodic structure. Final Energy: Non-dimensional strain energy in bay N’ of the optimised structure. No of Optimal Original Final Bay Reduction Bays, N Pattern Energy Energy No., N’ (dB) Q.=50 9 ULLLUUUUL 0.667E-1 0.297E-1 1 3.514 0.296E-1 1 3.528 10 ULLLUUUULU 0.540E 0 12 ULLLULUULULU 0.404E 0 0.297E-1 1 11.336 0.296E-1 1 11.351 40<Q<60 9 UUUUULULU 0.486E 0 0.710E-1 1 8.354 0.449E-i 1-2 10.344 10 ULULULULLL 0.606E 0 0.643E-1 1 9.743 0.45 IE-1 1-2 11.283 11 ULULULULLUU 0.456E 0 0.682E-1 I 8.252 0.425E-1 1-2 10.306 12 UUUIJUUUXJUUUL 0.332E 0 0.550E-1 2 7.808 0.412E-1 1-2 9.062 23<n<61 9 LLLLLLLLL 0.234E 0 0.203E 0 1 0.617 0.979E-1 1 3.784 n LLLLLLLLLL 0.200E 0 0.178E0 1 0.506 ■I 0.95 IE-1 1-2 3.228 11 UUUUUUUUULL 0.198E0 0.193E0 1 O.lll 0.910E-1 1-2 3.376 12 UUUUUOUIJULUU 0.314E0 0.182E0 I 2.369 0.803E-1 1-2 5.922 1123 Phase £ AUenualion Figure 1; A simply supported periodic Itlijsj kinelic.enorgy in bay 12, T, 1128 EFFECTS OF GEOMETRIC ASYMMETRY ON VIBRATIONAL POWER TRANSMISSION IN FRAMEWORKS J L Homer Department of Aeronautical and Automotive Engineering and Transport Studies, Loughborough University Loughborough, Leics , LEll 3TU, UK ABSTRACT Many sources, such as machines, are installed on supports, or frameworks, constmcted from beam-like members. It is desirable to be able to predict which wave types will be present at particular points in the support structure. By using the concept of vibrational power it is possible to compare the contributions from each wave type. Wave motion techniques are used to determine the expressions for vibrational power for each of the various wave types present. The results from the analysis show the amount of vibrational power carried by each wave type and the direction of propagation. Consideration is given to the effect on the vibrational power transmission of introducing misalignment of junctions in previously symmetric framework structures. By splitting a four beam junction in to, say, a pair of three beam junctions separated by a small distance, it is possible to establish the effects of separating the junctions on the various transmission paths. Unlike other techniques using vibrational power to analysis frameworks, the model keeps the contributions from each of the various wave types separate. This allows decisions to be made on the correct vibrational control techniques to be applied to the structure. INTRODUCTION When attempting to control vibration levels transmitted from a machine through the various connections to the structure upon which it is mounted, it is desirable to be able to identify and quantify the vibration paths in the stmcture. Often large machinery installations are installed on frameworks consisting of beam like members. These frameworks are then isolated from the main structure. Simple framework models are also used in the initial design stages of automotive body shell structures to determine dynamic responses. If the dominant transmission path in the framework is identified it is possible to reduce vibration levels by absorbing the mechanical energy along the propagation path in some convenient manner. By utilising the concept of vibrational power it is possible to quantitatively compare the wave type contributions to each transmission patL In order to predict vibrational power transmission in a framework, it is necessary to identify the wave amplitude reflection and transmission coefficients for each joint in the structure. Lee and Kolsky [1] investigated the effects of longitudinal wave impingement on a junction of arbitrary angle between two rods. Similarly Doyle and Kamle [2] examined the wave amplitudes resulting from a flexural wave impinging on the junction between two beams. By using the reflection and transmission coefficients for different joints, it is possible to predict the vibrational power associated with flexural and longitudinal waves in each section of the 1129 framework. Previous investigations [3,4] have considered the effects of bends and junctions in infinite beams. This work was extended to consider the finite members which constitute frameworks [5]. Unlike other techniques [6, 7] utilising energy techniques to analyse frame-works, the technique produces power distributions for each wave type present in the structure. By comparing the results for each wave type, it is possible to apply the correct methods of vibration control. The technique is used to investigate the effect of geometric asymmetry on the vibration transmission, due to steady state sinosodial excitation, in a framework structure similar to, say, those used in the automotive industry (figure 1). By splitting a four beam junction into a pair of three beam junctions separated by a known distance, it is possible to establish the effect of junction separation on the dominant transmission paths. The investigation presented is limited to one dimensional bending waves and compressive waves only propagating in the structure. To consider the addition of other wave types ie. torsional waves and bending waves in the other plane, the analysis presented here for the junctions should be extended as indicated by Gibbs and Tattersall [3]. TRANSMITTED POWER IN A UNIFORM BEAM For flexural wave motion, consider a section of a uniform beam carrying a propagating flexural wave. Two loads act on this beam element, the shear force and the bending moment. It is assumed that the flexural wave can be described by using Euler-Bemoulli beam theory, so that the displacement can be expressed as W(x,t) = Af sin (cot-kfx), the shear force acting on a section as S = Eia3W/ax3, and the bending moment on the section as B = El 02 W/ax2. Then the instantaneous rate of working X at the cross-section is given by the sum of two terms (negative sign merely due to sign convention). 0t 0x0t 0X^ 0X^ 3x0t The time averaged power {P)f = (1/T) I X dt then is given by (P)f = Elkf coAf I ( 1 ) For longitudinal wave motion consider a section of a uniform beam with a longitudinal wave propagating through the beam U (x, t) = Ai sin ( 0 )t-kix) 1130 The instantaneous rate of working X is then X= -EA(au/ax)u and the time averaged power is Xdt = rEA0)kiA| ( 2 ) If dissipation is present in the structure, the modulus of elasticity may be considered to be a complex quantity = E(l + iTi) where represents the loss factor of the material, present due to inherent material damping. The displacement of a beam at a distance x from the source, due to flexural wave motion may now be considered to be, assuming that material damping is small. 1 ^ -kri- W = Afe 4 and the resulting time averaged power is given by (P}f = EIcok^e-'‘'’i2 Af (3) The above reduces to equation (1) at the source. Similarly, the displacement of beam, due to longitudinal wave motion may be considered to be T ^ -kiTi- U = Aie ^ sin(cot-kix) and the resulting time averaged longitudinal power may be rewritten as (P),=iEAcok,e-‘''^’‘A? (4) WAVE TRANSMISSION THROUGH A MULTI BRANCH JUNCTION Consider a four branch junction as shown in figure 2. Assuming only flexural and longitudinal waves propagating in the structure, the displacements of Arm 1 will be, where A 4 represents the impinging flexural wave arriving from infinity. Wi(x,t) = (Aie^f’"" + A 3 e‘^fi'' + A 4 e e‘“^ (5) 1131 ( 6 ) U,(x,t) = (Aae''‘‘i’‘)e‘“‘ Similarly for arms 2 to 4 the displacement will be, where ^ cos 0 n and n is the beam number W„(v„,t) = )e‘“‘ (V) ( 8 ) Here A 3 , A 4 , 64 ^ are travelling flexural wave amplitudes', Ai and B 2 n are near field wave amplitudes and Aa and are travelling longitudinal wave amplitudes. In previous work [2] in this field a theoretical model was used in which it was assumed that the junction between the beams was a rigid mass. The mass or joint is modelled here as a section of a cylinder. This represents the physical shape of most joints in practical systems. It has been shown [4] that the joint mass has an insignificant effect on the reflected and transmitted power for the range of values used in this work. The joint mass Mj = pjTtL^J^/ 4, and the moment of inertia of the joint isIj=ML2/8. By considering the conditions for continuity and equilibrium at the beam junction the following expressions may be written. For each arm For continuity of longitudinal displacement L 3W Ui=Un COsen-WnSinen + -^^sinen For continuity of flexural displacement W,=u„sinen+W„cos0„-| |^(l + cose„) For continuity of slope 8W; ^ awn 1132 For the junction Equilibrium of forces ' ' 2 ' ' 3x2 J -vfp fax L32 w„ 11 I aVS 2 3 v|;^ El Ai ^ + Mj ^ = i[E„ A„^cose„ +E„ I„^^sine„ 3x J dt^ il^ 3V„ 3< E T a^W, 3 r„, L3W| El Ii —t^ + M; —T Wi- ——— -^ -I- iVi; —y 1 “ “ ^T" 3x2 2 3x n ;^TI ^ W = Z En Ajj- ”Sin6n-EnIn 2 *^ COS0n 1 I 5¥n WAVE MOTION AT A FORCED OR FREE END As indicated in figure 1, the framework has one forced end and one free end. Assuming the structure is only excited by a transverse harmonic force, the boundary conditions are as follows: at the forced end E« = Pe^«t 3x2 EI^ = 0 3x2 EA^ = 0 Similarly at the free end the above boundary conditions apply with the exception that EI^ = 0 3x2 1133 POWER TRANSMISSION THROUGH A FRAMEWORK The structure shown in figure 1 consits of one four-beam junction, two three- beam junctions and four two-beam junctions. From the equations detailed in the above two sections, it is possible to construct matrices of continuity and equilibrium equations for sub structures. These may be combined to obtain the overall matrix for the system. For the framework shown in figure 1, the size of the overall matrix is 60x60. This matrix may be solved to obtain the sixty unknown wave amplitude coefficients from which time averaged transmitted power for each beam may be calculated using equations (3) and (4). Normalised nett vibrational power is then calculated at the centre of each beam constituting the structure. Nett vibrational power may be considered to be the difference between power flowing in the positive direction and power flowing in the negative direction for each wave type. Normalised nett power is considered to be nett power divided by total input power. The input power to a structure may be calculated from the following expression [8] Input Power = ^ IFIIVI cos0 where 0 is the phase angle between the applied force and the velocity of the structure at the forcing position. Figures 3-6 show the nett normalised power in each arm of a framework structure over the frequency range 0-lkHz excited by IN force, whose material and geometric properties are given in Appendix 2. For the results shown, angle 1 is 45® and angle x is 40® (or the ratio = 0.89) and L = y = 0.1m. Using these parameters the ratio of the length of beam No.6 to beam No.4 is 0.12. The predicted flexural power is shown in figures 3 and 4 and from these it can be seen that the dominant transmission paths are arms 1 and 5, the forced and free arms. The transmitted power in arm 10 is next dominant and comparable to arm 5 in the region 0-600Hz. The response for ail other arms are small, typically less than 5% of input power, with, as would be expected, arms 2 and 9 being approximately identical in transmission properties. Figures 5 and 6 show the nett normalised power for the longitudinal waves in the structure. As the frequency range of interest corresponds to a flexural Helmoltz number of 1 to 5 with L being the reference length, the conversion of power from flexural to longitudinal waves is minimal. From the figures it can be seen that beams 1, 5, 6 and 10 have identical transmission characteristics, which would be expected at such large longitudinal wavelengths. Significant longitudinal power is only observed in arms 3 and 8 in the frequency region 200-300Hz. This frequency region coincides with a drop in the flexural power due to the structure being at resonance in that region. It should be noted that power transmitted through arms 3 and 8 has travelled through two junctions. 1134 EFFECT OF GEOMETRIC ASYMMETRY By altering the ratio of angle 6i to angle 0x it is possible to alter the length of beam 6 and hence move a pair of three arm junctions further or closer apart. From the discussion in the previous section, it was seen, for the structure under investigation, that the dominant flexural path, not surprisingly, is through the centre of the structure, whilst the peaks in longitudinal power occur in beams 3 and 8. Thus 0x was varied and the effect on transmission in the dominant paths noted. Figures 7-9 show flexural power for arms 5 and 10 and longitudinal power for arm 8 for four values of 0x. The values chosen were 36°, 38.25°, 40° and 42.75° which are equivalent respectively to 0x over 0] ratios of 80%, 85%, 90% and 95%. Thus as 0x increases, the structure moves to being symmetrical in nature. From figure 7, the increase in junction separation decreases power in the frequency region 0-500Hz and increases it in the region 500-lkHz. In beam 10 (figure 10) the effect on the flexural power is reversed with increase in junction separation leading to increased power below 500Hz and decreased power above 500Hz. It should also be noted that increased junction separation has little effect on the power below 250Hz. The effect was also noted on all other beams which had both ends connected to a joint. It may be concluded that at long flexural wavelengths the junction separation has little effect with the impedance mis-match at the junctions being the important criteria to effect transmission. It should also be noted that increasing power in one arm ie. 5, causes a decrease in arms ie. 10, connected to it. An example of the effect of junction separation on longitudinal power is shown in figure 9. This shows nett normalised longitudinal power for arm 8 for the same variation in 0x. Again minimal effect is seen at low frequencies, with increased junction separation having different effects in different frequency region. Increasing junction separation has little effect on the region between 200 and 300Hz when the longitudinal power was dominant. This would be caused by the junction separation having little effect on the structures flexural natural frequencies. Only by shifting those would the peaks in longitudinal power by shifted in frequency. CONCLUSIONS Results are presented for normalised nett time average vibrational power for a framework structure. The geometric symmetry of the structure is broken by allowing one angle to decrease in value. The effects of varying the angle change by up to 20% of its original value are investigated. Although the results presented are for one example only, highlighted are the fact that decreases in power in one part of the structure result in increases in power in another part. Also shown was the effect of splitting a junction in to a pair of junctions is minimal at low frequencies, or long wavelengths. From the results of the analysis it is possible to establish frequencies and positions for minimum power on the structure. Other configurations of framework structure may be analysed by applying the equations presented. 1135 REFERENCES 1. J. P. LEE. and H. KOLSKY 1972 Journal of Applied Mechanics 39, 809- 813. The generation of stress pulses at the junction of two non-collinear rods. 2. J. F. DOYLE and S. KAMLE 1987 Journal of Applied Mechanics 54, 136-140. An experimental study of the reflection and transmission of flexural waves at an arbitrary T-Joint. 3. B. M. GIBBS and J. D. TATTERS ALL 1987 Journal of Vibration, Acoustics, Stress and Reliability in Design, 109, 348-355. Vibrational energy transmission and mode conversion at a corner junction of square section rods. 4. J. L. HORNER and R. G. WHITE 1991 Journal of Sound and Vibration 147, 87-103. Prediction of vibrational power transmission through bends and joints in beam-like structures. 5. J. L. HORNER 1994 Proceedings of the 5th International Conference on Recent Advances in Structural Dynamics, SOUTHAMPTON UK, 450- 459. Analysis of vibrational power transmission in framework structures. 6. P. E. CHO and R. J. BERNHARD 1993 Proceedings of the 4th International Congress on Intensity Techniques, SENLIS, France, 347- 354. A simple method for predicting energy flow distributions in frame structures. 7. M. BESHARA and A. J. KEANE 1996 Proceedings of Inter-Noise '96, LIVERPOOL, UK 2957-2962. Energy flows in beam networks with complient joints. 8. R.J.PINNINGTON and R.G.WHITE 1981 Journal of Sound and Vibration 75, 179-197. Power flow through machine isolators to resonant and non- resonant beams. APPENDIX 1 - NOTATION A - Cross sectional area Q - Axial force Af - Amplitude of flexural wave s - Shear force Ai - Amplitude of longitudinal wave T - Time period B - Bending moment t - Time E - Young’s modulus U - Displacement due to E* - Complex Young’s modulus longitudinal wave motion F - Excitation force V - Velocity I - Moment of inertia w - Displacement due to flexural wave motion Ij Jw - Moment of inertia of joint X - Instantaneous rate of working - Joint width X - Distance - Loss factor kf - Flexural wave number 0n - Angle of Arm n ki - Longitudinal wave number Pj - Joint density L - Joint length M - Moment force Mj n P - Joint mass <}> Phase angle - Beam number - Transverse force ¥n - Distance along Arm n 1136 <P>f - Time averaged flexural power co - Frequency (rad/s) <P >1 - Time averaged longitudinal power APPENDIX 2 - MODEL PROPERTIES 33mm 6mm 5GN/m2 1180kg/m3 0.001 1 Beam Breadth Beam Depth Youngs Modulus Density Loss factor Figure 1: Framework Structure iNett Normalised Power Nett Normalised Power Figure 5: Longitudinal Power - Beams 1-5 (Beam 1-, Beam 2., Beam 3 . . Beam 4-, Beam 5 Figure 6: Longitudinal Power - Beams 6-10 (Beam 6-, Beam 7., Beam 8 . . Beam 9-, Beam 10 1139 THE INFLUENCE OF THE DISSff ATION LAYER ON ENERGY FLOW IN PLATE CONNECTIONS Marek Iwaniec, Ryszard Panuszka Technical University of Mining and Metallurgy, Structural Acoustics and Intelligent Materials Group 30-059 Cracow, al. Mickiewicza 30, Poland 1. Introduction Dynamic behaviour of mechanical strucmres may be modelled on the basis of and with the help of mathematical apparams used in Statistical Energy Analysis (SEA) [5] The method is especially useful to calculate the statistical approach vibroacoustical energy flow in middle and high frequency range. With the help of a few parameters, such'as- modal density, damping loss factor, coupling loss factor and the value of input power, building linear equations set it is possible to describe the flow of vibroacoustical energy in a complicated stmcture. There is also a possibility of quick estimation of the influence of constmction method on the vibroacoustical parameters of the whole set. In the following work an exemplary application of one of the most frequently used software for calculating the flow of acoustic energy has been presented- AutoSEA programme [1]. The aim of the work is practical modelling of vibroacoustical energy flows through screw-connection of two plates and comparing quantity results with experimental (outcome) measurements. Equivalent coupling loss factor has been calculated for a group of mumally combined elements constimting a construction fragment. A comparison between the measured results and the value of coupling loss factor in linear joint (e.g. in welded one) has also been made. Using the method of fmite elements, the influence of rubber separator thickness on the value of the first several frequencies of free vibrations has been computed as well. 2. Physical model of plate connection A connection of two perpendicular plates has been chosen for modelling the flow of vibroacoustical energy in mechanical joints. Connection diagram is presented in Figure 1. On the length of common edge the plates has been joined with anglesteel by screws. A rubber separator (4) has been placed between the excited plate and the anglesteel leg (3). 1143 Modelled stmcture 3. SEA model In order to carry out the vibroacoustical analysis of the system using Statistical Energy Analysis a model of the examined strucmre has been built. It has been assumed that in every element of the construction only flexural waves propagate. Every plate and the rubber layer have been modelled with just one appropriately chosen subsystem. The anglesteel, however, has been modelled as a continuous connection of two plates having the dimensions which correspond to the anglesteel legs the plates themself being set at the right angle. Fig. 2. SEA model of a system Fig- 3. Modal densities of the subsystems 1144 Using the SEA method we are able to describe the flow of the vibroacousdcal energy in middle and high frequencies with an algebraic equation set. The exMined system consists of five simple subsystems, of which only one is exited to vibration with applied force. The flow of vibroacoustical energy m the model presented is depicted with the following equation: ^ 1 tot -^21 0 0 12 ^ 2 tot 32 0 0 “'•123 ^ 3 tot -‘n43 0 0 “''134 h 4 tot 0 0 0 "'’145 0 tot w, 0 ^2 tot 0) 0 0 • ^3 tot = 0 -^54 ^4 tot 0 ^ 5 tot ^5 tot 0 ^ ^ 4 - is a total coefficient of energy loss for every subsystem, rj’. '"' - intemarioss factor of the subsystem, ri.. - coupling loss factor between subsystems, E- - the mean vibrational energy in Af frequency band in i-subsystem, W - the input power carried into i-subsystem from outside. To determine the elements of the coefficients matrix in eq. 1 it is necessary to know [1,4,6] coupling loss factors (CLF) between structural subsystems and damping In the SEA model in question transmission of the acoustic energy occurs in two tvpes of connections between; the plate and the beam (the point joint of the beam which is parallel to the edge of the plate, (transfers flexural waves), . . „ two plates (linear connections and point joints transferring flexural waves). The coupling loss factor between the plate and the beam which vibrate in the flexural way (in the case of the point joint), is defined with following equation [1]. 1.75c^ ( 2 ) where: c - is the speed of flexural wave, T - transmission factor, Q - the number of point connections, Gj - angular frequency, A - the surface of the plate. 1145 The speed of the flexural wave in the first plate c„ can be calculated in the following way: (3) In the model under examination formula (2) defines coupling loss factors between the anglesteel leg (2) and the beam (4) - (factors 1 , 3 . and tiJ or the beam (4) and the plate (5) - rj 45 and 7 / 54 . The flow of energy between two plates (which are connected at the right angle and which vibrate in flexural way) is defined with the following formula in the case of linear connection: where: 1 - is the length of the connection. With above formula it is possible to describe the flow of energy through correctly made welded joints of plates or, for example, through bent plates. In the system presented the factor determines the flow of energy between anglesteel legs ( 2 ) and (3) The coupling loss factor between two plates with a point joint is described by the following formula: iiL^ 3 (o.X, (5) This type of connection occurs between the plate representing the anglesteel leg and the plate (5). ^ . After defining the value of factors matrix in the first equation it is possible to specify the ratio of the vibroacoustical energy gathered in plate (1) and (5). _ (n 2tof^ 3t0t~ ^ 24^ 32 ) 4fot^ 5tnt~ ^ 54^ 45^ ~ 34^ 43^ 5tot (5) £5 ^ 21 ^ 32 '^ 43^ 54 The damping loss factor is important parameter of every subsystem. For steel plates used in the experiment the value of the damping loss factor have been measured experimentally with the decay method. The results of the measurements have been presented in figure 4. The frequency characteristic of rubber damping has been shown in figure 5. 1146 DLF of steel [-] frequency [Hz] frequency [Hz] Fig. 4. Damping loss factor of steel Fig. 5. Damping loss factor of rubber 4. Experimental research Experimental investigations have been carried out for connections made with the use of rubber separator (elastic layer) of 50° Shore hardness. The connection was build up of two identical, perpendicular plates connected each to another using the anglesteel and the elastic rubber layer. These are the properties and material parameters of individual elements: plates: - material constructional steel (St3); - dimensions: 500 * 500 * 2.2 mm; - Young modulus: 2,1 10'^ Pa. anglesteel: - material: constructional steel (St3); - dimensions: L 40 40 2.2 mm; - Young modulus: 2,1 10'‘ Pa elastic layer: - material: rubber 50° Shore - dimensions: 500 40 mm - thickness: 2, 3, 4, 5, 6 mm To avoid the loss of mechanical energy in the environment, during measure¬ ments the construction was suspended to the supporting frame with three weightless strings in such way that only rigid body motions in the plane perpendicular to the plate surface can occur. The excitation of the wide-band type with constant power spectral density was applied in the symmetry axis of the plate (5) about 20 mm below the upper edge (Figure 1). During the experiment the distribution of vibrating velocities on the plate surface was obtain by non contact method using laser-vibrometry. 1147 5. FEM model Vibration of modelled structure in low frequency has been analyzed by toe Finite Element Method (FEM). Several FEM models, was build in order to consider L valid thickness of the mbber layer. The mbber thickness has been from 0 mm to 6 mm. The dimensions and material parameters of the plates and the LSesteel was constant. The structure was fixed in four comers. Calc^auons were mfde for a division of the strucmre into 608 elements of type BRI^8. The mfluence of the rubber thickness on the eigenfrequencies was remarkable. The results for first 15 eigenfrequencies are presented in the table 1. In fig. was shown also the changes of the value of natural frequencies in comparison with the natural frequencies of strucmre without applying the mbber layer. table 1. Namral frequencies of connection LaDlc i. i>aLutai ai. — - 1 Natural frequencies (Hz) for various rubb ■ ■. - - • er layer tlrickness 0 nun 2 nun 3 mni 4 nun 5 nun 127.8 123.1 121.5 119.4 117.5 297.5 235.8 224.6 212.6 204.6 507.6 432.5 391.3 357.1 330.0 1003.2 566.9 525.7 503.7 491.9 1282.3 807.1 768.9 733.5 712.2 1900.5 1185.2 1169.4 1123.2 1070.2 2386.7 1570.2 1483.6 1376.2 1304.6 3272.3 1851.6 1634.2 1531.6 1490.8 3603.4 2303.1 2263.4 2167.8 2147.3 4263.3 2611.2 2539.5 2496.9 2463.7 4759.5 3192.7 3515 3110.1 2913.2 5438.4 3926.7 3874.6 3626.4 2937.0 7627.4 4110.1 4105.5 3639.9 3026.6 7691.6 5070.7 5137.0 3670.5 3111.2 11543.2 5547.1 5503.9 3721.9 3181.2 The decreasing of the absolute values of the natural frequences is observed according to increasing of the thickness of the rubber layer. Beginning from die third of °ib adons of the system the decrease of the natural frequencies is almost constant for the "layer 2 mm or 3 mm and is continuously decreasing for rubber layer 4-6 mm (See fig. 6). 1148 Fig. 6. Changes of natural frequencies 6. Comparison with experimental results The equivalent coupling loss factor defining the energy flow between the plates has been determined experimentally [31 for a model consisting of two subsystems; two plates. The coupling loss factor in such two element model may be specified by the following equation: 1 zast. ^2 ^1101 N, E2.0C - 1 (7) The quotient of plate energies E,/E, in a two-element model is relevant to the quotient of energy in the first and fifth subsystem (E./E,) in the five element model presented m figure 0 the value of these quotient is defined with the equation (6) In the picture we have presented the values of the equivalent coupling loss factor in the connection. Individual points in the diagram show the results obtained experimentally. The values received in computer simulation have been presented as a continuous diagram. The upper curve shows the values of the coupling loss factor in the joint before the application of the rubber layer. In the frequency range above ca. 125 Hz we have received a very good comparison of experimental results and computer simulation results performed with the AutoSEA software At the frequency of about 200Hz there occurs a local minimum of the equivalent coupling loss factor between the plates. The value of the minimum is essentially influenced by the value of the rubber damping loss factor. The frequency (with the minimum CLF) is strongly influenced by the peak frequency of the rubber damptng curve. 1149 o.ooo2ii- symulation experiment welded plates 16' ‘ '40' ' ’lOO 250 630 1600 4000 frequency [Hz] Fig.7 Equivalent CLF factor in the joint 7. Conclusion A way of modelling the vibroacoustical energy flow with the help of SEA method has been presented in the work. We have examined the screw connection of two plates, where a rubber elastic layer has been applied, A comparison has also been made between the results of computer simulation of the mechanical energy flow with SEA method and the experimental results, and thus we have noticed the good correlation, especially as far as middle frequencies The joint modification through introduction of the rubber separator has a remarkable impact on the acoustic energy flow: , ^ - The application of the elastic layer in the Joint in question lowers the value ot eauivalent coupling loss factor in the whole frequency range. - the value of rubber damping factor has most significant influence on the acoustic eneroy flow through connection in middle frequencies range: “ The minimum value of the equivalent coupling loss factor in a joint is essentially influenced by the rubber separator damping loss factor. * The frequency of minimum CLF occurrence is strongly influenced by the peak frequency of the rubber damping curve. The increasing thickness of the rubber layer produces on decreasing natural frequencies of the structure. 8. Bibliography: III AutoSEA - User Guide Vibro-Acoustic Sciences Limited 1992. 121 Fahy F.J. Sound and Structural Vibration; Radiation, Transmission and Response Academic Pres nTkirtuin J Smals N. Panuszka R. "Method of estimating the coupling loss factor for a set of nlates" Mechanic, Technical University of Mining and Metallurgy, Cracow, 10,1991. [^41 Lalor, N.: The evaluation of SEA Coupling Loss Factors. Proc. V School Energy Methods in Vibroacoustics" - Supplement, Krakow-Zakopane 1996. , . „ „ [51 Lyon R., DeJong R.; Theory and Application of Statistical Energy Analysis. Butterworth- Heinemann, Boston, 1995. 1150 Variation Analysis on Coupling Loss Factor due to the Third Coupled Subsystem in Statistical Energy Analysis Hongbing Du Fook Fah Yap School of Mechanical & Production Engineering Nanyang Technological University Singapore 639798 Abstract Statistical Energy Analysis (SEA) is potentially a powerful method for analyzing vibration problems of complex systems, especially at high frequen¬ cies. An impoitant parameter in SEA modeling is the coupling loss factor which is usually obtained analytically based on a system with only two cou¬ pled elements. Whether the coupling loss factor obtained in the classical way is applicable to a practical problem, which normally comprises of more than two elements, is of importance to the success of SEA. In this paper, the varia¬ tion of coupling loss factor between two subsystems due to the presence of a third coupled subsystem is investigated. It is shown that the degree to which the coupling loss factor is affected depends on how strong the third subsystem is coupled. It also depends on the distribution of the modes in the coupled sub¬ systems. This kind of effect will diminish when the damping is high, subsys¬ tems are reverberant, or ensemble-average is considered, but not for individual cases. 1 Introduction SEA is potentially a powerful method for analyzing vibration and acoustic problems of complex systems, especially at high frequencies, because of the simplicity of its equations compared to other deterministic analysis techniques. SEA models a system in terms of interconnected subsystems. The coupling parameter between any two subsystems is characterized by a coupling loss factor. If the coupling loss factors and internal (damping) loss factors of all subsystems are known, the power balance equation (e.g., see [1]) for each subsystem can be established. From this set of equations, SEA predicts the system response (due to certain types of excitation) in terms of the average energy of every subsystem. The energy can in turn be related to other response quantities such as mean velocity or strain. 1151 Historically, the SEA power balance equations were initially derived from an analysis of two coupled oscillators [2,3]. It has been shown that the energy flow between them is directly proportional to the difference in their uncoupled modal energies. The theory has then been extended to systems with multi-coupled sub¬ systems (e.g., [4]). Strictly this extension is only applicable if certain assumptions are justified [4,5]. Also the new concept of indirect coupling loss factor, which is used to represent the energy flow proportionality between the indirectly coupled subsystems, is also introduced. In practice, the indirect coupling loss factors are normally ignored in SEA ap¬ plications because they are very difficult to determine analytically. Only coupling loss factors between directly coupled substructures are considered. Some analy¬ ses [6-8] have shown that this approximation may lead to significant errors in the predicted results if certain conditions are not met in the system. These conditions include not only the well known requirement of weak coupling between subsystems (e.g., see [6]), but also others, such as given by Langley that the response in each element must be reverberant [7]; and by Kean that there should be no dominant modes (peaks) inside the frequency-averaging band [8]. However, it is usually dif¬ ficult to know whether these conditions are satisfied for a particular system. In fact the above mentioned conditions do not always hold for practical engineering cases. On this point of view, the importance of a coupling loss factor for describing the coupling between indirectly coupled subsystems are to be further examined. A related question is whether the coupling loss factors obtained from the system with only two subsystems can still be applied when other subsystems are present. Generally, the coupling loss factor is sensitive to the amount of overlap between the modes of the two coupled subsystems. When additional subsystems are cou¬ pled to the original two-subsystem model, the mode distributions of the originally coupled two subsystems will be affected. The change of mode distributions will fur¬ ther affect the modal overlap between the coupled two subsystems and finally the coupling loss factor between them. However, general estimation methods for cou¬ pling loss factor assume that the coupling parameters between two subsystems are not affected much by the presence of the other subsystems. Therefore the conven¬ tional approaches of deriving coupling loss factor are mostly based on consideration of a two-subsystem model only. One method is the wave approach, by which the coupling loss factor used in the SEA applications are derived analytically from aver¬ aged transmission factors of waves that are transmitted through a Junction between semi-infinite subsystems. This method only takes into account local properties at the joints and sometimes may be inaccurate. Recent research [1,7,9-11] based on the model with two-coupled subsystems has shown that the coupling parameter does depend on other system properties, such as damping loss factor, etc. It can be argued that, if there is a third coupled subsystem, the coupling parameters between the first two subsystems will also depend on the energy flow to the third subsys¬ tem. Therefore, from a practical point of view, the coupling loss factor estimated for two-coupled subsystems, ignoring the indirectly coupled subsystems, can only be of approximate value. 1152 In this paper, the variation of coupling loss factor between two subsystems due to the presence of a third coupled subsystem is studied. In the following sections, the coupling loss factor is firstly expressed in terms of global mobility functions. The exact solution of mobility functions is only for simple structures. However, for general structures, it can be obtained by Finite Element Analysis (FEA) [10,11]. The coupling loss factors obtained respectively in the cases with and without the third subsystem in the model are compared for two particular system configura¬ tions, respectively. The system used in this investigation is one-dimensional simply supported beanis coupled in series by rotational springs. By varying the spring stiff¬ ness, the strength of the coupling between the second and the third subsystems can be changed. It is shown that the effect of the third coupled subsystem on the cou¬ pling loss factor between the first two coupled subsystems depends on how strong the third subsystem is coupled. For each individual case, it is also shown that this kind of effect may be positive or negative, depending on the distribution of modes in the coupled subsystems. 2 Coupling Loss Factor by Global Modal Approach In this section, a modal method is used to derive coupling loss factor in a sys¬ tem with any number of coupled subsystems. The result is then simplified for two cases: (1) a three-subsystem model; (2) a two-subsystem model which is simply substructured from the previous three-subsystem model by disconnected the third subsystem. For a linear system which consists of N coupled subsystems, if “rain-on-the- roof” excitation [10] is assumed to be applied to each subsystem in turn, the corre¬ sponding response energy can be expressed as I I drdsdu, (1) sith.'ii/sl.Kmi Mii.bsy.'it.emj where /?/; is the total time-averaged response energy of the subsystem i. due to the excitation on the subsystem j, is the transfer mobility function be¬ tween the response points r and the excitation point H is the averaging range of frequency, u.', in and S are the mass density and the power spectral density of excitation. The input power due to the excitation is given by I RelH(s,.i,L 0 )]d.sdu (2) Q sahsyale'inj where is the real part of the point mobility at the position simplicity, two terms, a/, and are defined as a,, = i>, = ^ = Al m;S j Ik \ H[r, s,uj)\~ dr ds cl.in n sii.bsy.'iUjvii /iubfiyslenij lle[H{.^,s,ij)] dsd.u: n subsyslr;mj For (3) (4) 1153 The mobility function, H{r,.s,uj), is to be expressed in terms of the global modes of the system, which can be obtained by Finite Element Analysis (FEA). By the principle of reciprocity of the mobility function, the relation of aij = a ji always holds regardless of the strength of coupling and the magnitude of input power if the excitation is “rain-on-the-roofTheoretically applying the Power Injection Method [12] we can obtain the SEA equation as n = [77]E (5) where H = {Hi, ila, - • • , and E = {E[, E-z, - - - , E^r}'^. The SEA loss factor matrix [ 77 ] is iVl + i.^2 (7?yV + ■niiau niiCLiisr bi b-z bN ■nua-ii ■nizCL-zN bi bo bjV 77?.;Va,Yi •m.A/a/VY ^1 Ih b^! = — B A“'M- UJr where, //; is the internal loss factor for subsystem i., v/,, is the coupling loss factor from subsystem -I. to subsystem j, ujc is the central frequency of the averaging band n. A = [a,,] From equation ( 6 ), the reciprocity principle of the coupling loss factors can be easily seen, due to aij = aji. 1 ''■bi , M = L ’-J Vij n?.; b j ni jbj/ (cu^/i / 2) i]ji niibi ■niibi/iujc'^l'l) Where, ( 77 ?.; 6 ,:)/(u,v 7 r/ 2 ) is the averaged real part of point mobility [4] and can be regarded as the generalized modal density of the subsystem i. Assuming weak coupling and light damping, it approximately equals to the classical definition of modal density [13]. Therefore, the relation given by equation (7) also reduces to the classical reciprocity principle. 1154 2.1 Substructured two-subsystem model Figure 1: A general SEA model with three coupled subsystems in series Consider a whole system with three substructures coupled in series as shown in Figure 1. If subsystem 3 is removed, the coupling loss factors between subsystems 1 and 2 are given in the equation biCL22 bi(Li-2 "h'^712 — ^?2L _ ^ n?. i(fi 1 [fi 22 — '^7.2(^11^22 — <'^i 2 ^^' 2 l) — 7/12 772 + 7/21 u-V _ ^20.21 _ _ ^2(111 _ 777 i{an<^f 22 “ <^^ 12 f'' 2 l) 7772(^110-22 — O12O21) r bj ^to -12 1 (8) ^ J_ 777 ifli| 7772011022 62O21 ^2 7771O11O22 777.2 0 22 The approximation in the above equation is due to 011022 » 012 O 21 when the cou¬ pling is weak. Manipulating equation (8) with or without using the approximation both can work out the coupling loss factors 7/12 and 7/21 as (9) r 61 6-, (777 1 I ) (77 72 62 ) - 7/2 -77 07^-777-101 1 07^777.2022 ! , ^2 . f. *^1 77 7 20'7- 777 [Oi "07c7772022 07^-777-1 011 ( 10 ) The equations are true regardless of the strength of the coupling. It can be seen that 7/12 and 7/21 depend on the values of the three terms 777 .,+/, 7 // and bij{u:^ni-,au). The first two are the generalized modal density and the internal loss factor, or in combi¬ nation equivalent to modal overlap factor. The third one, by noting the definitions of 6; and an, is the ratio of input power to response energy for the directly excited subsystem, i.e., the total loss factor of subsystem i. From equation (8), this term can be approximately expressed as Total loss factor of subsystem i b; miLOcCLii m + 5I'+-.7 ( 11 ) 1155 In the classical wave approach, where semi-infinite subsystems are assumed, the total internal loss factors becomes ■n total, I = m + Vn and l]iot.al,2 = 112 + vTi (^2) where is the classical coupling loss factor. Substituting equation (12) into equa¬ tions ( 9 ) and ( 10 ), i]ij reduces to the classical iiff, which only depends on the local properties at the joints rather than other properties of the system, such as damping. If the subsystem modal parameters are used to evaluate the term, bil{ujcmiaii), then the total loss factor is Just the internal loss factor of the subsystem and the cou¬ pling loss factor is equal to zero. This is reasonable because using the uncoupled modal parameters instead of the coupled modal parameters is actually equivalent to removing the coupling between two subsystems. However, for finite system where the assumption of semi-infinity is not justified, there will be no immediate simplication for equations (9) and (10). Numerically, FEA can be employed to obtain the global modes and then the coupling loss factor can be calculated [ 10 , 11 ]. 2.2 Full three-subsystem model Instead of substructuring, consider the three-subsystem model as a whole system, shown in figure 1. Now the order of equation ( 6 ) is reduced to 3. With the global modal parameters obtained from FEA, the coupling loss factors can be directly evaluated. However, when the coupling between subsystems is weak, the order- reduced equation ( 6 ) is still able to be simplified. Matrix A may be alternatively expressed in the form of ■ a, L 0 0 ‘ ■ 0 «L2 0 ■ 0 0 «13 ' A - 0 (t-ll 0 -f a\2 0 d.23 + 0 0 0 0 0 «33 . 0 ((•23 0 . . (l.[3 0 0 where the terms on the right side are sequentially defined as Ai, A 2 and A 3 . Under the assumption of weak coupling, the non-zero entries in Ai, A 2 and A 3 will be of the order O(t^), (9(e^) and respectively [14], The inverse of matrix A may be approximately written as = Ai”^ — Ai“^A2Ai”^ — Ai"^A3Ai ^ 4 -Ai ^A2Ai ^A2Ai + ••• (14) As an approximation, substituting only the first three terms in equation (14) into equation ( 6 ) gives [tj] ^ l/uv- />■ bo f t’12 rn I f( 11 a 2 2 b3(l-\3 ni[a\ KM.3 ■ni2a 11 (1-22 b2 1112(1.22 63^23 1112 ( 1 . 22^(33 bi(t.\3 in3(l\\(l.33 bid. 13 in.3Ct.2'i((33 63 ~ 1((3((33 (15) 1156 Generally, under the condition of weak coupling, the indirect coupling loss factors, ■ihs and 7731 , are much smaller than the direct coupling loss factors and the internal loss factors [5,7,15]. The diagonal elements in [rj] can therefore be approximated to the sum of internal loss factor and direct loss factor. It can be shown from equation (15) that equation (11) remains valid for three coupled subsystems. But in the three- subsystem case, the term, b,|{uJcm.^au), is to be evaluated by using the global modes of the three-subsystem model. 3 Numerical Examples and Variation Analysis In this section, two examples with different configurations are used to show the vari¬ ation of coupling loss factor due to the presence of a third coupled subsystem. The coupling loss factor of the two-subsystem model is evaluated by using equations (9) and (10). For the three-subsystem model, equations ( 6 ) and (15) are used. It can be shown that both equations ( 6 ) and (15) give the same results as the couplings are weak. 3.1 Structural details and SEA model Ki beam I (TJT] - (a) ^ -Zi. beam 1 rnn beam 2 (TTH ^ (b) ^ ^ -A A- Figure 2; A structural model comprising of three beams To begin example calculations, consider initially a two-subsystem model (figure 2(a)) which is two thin beams coupled through a rotational spring. The group of flexural vibration modes of each beam are taken as a SEA subsystem. The spring provides weak coupling between them where only rotational moment is transmitted. When beam 3 is connected at the free end of beam 2 to the original two-beam model, a three-subsystem model is formed (figure 2(b)). The specifications for the three beams are given in table 1. The spring stiffness, A' 2 , is adjustable in order to look into the significance of the effect of the third subsystem. There are two cases where the length of beam 2 is: (i) L 2 - 1.0?72; (ii) L 2 = l-lm. The spring constants at the joints are chosen to be weak enough to ensure that:(a) the coupling loss factor is much smaller than the internal loss factor; (b) the indirect coupling loss factor is much smaller than the direct coupling loss factor. In the global modal approach (see section 2 ), the modes of two-subsystem model and three-subsystem model are obtained from FEA. In numerical simulation, the 1157 Table 1: The specifications of the three beams BEAM 1 2 3 length (m) 2.0 1.0 & 1.1 0.7 width (mm) 4 Thickness (mm) 2 Density (Kg/m^) 7890 Young’s Modulus (N/m'^) 196E+9 Poisson Ratio 0.29 spring constant, A*i (Nm/rad) 1.0 central frequency is 200Hz and the averaging band is selected as 100 ~ 300Hz. In order to take into account the contribution from the modes out of the band, all the modes up to 500Hz are extracted for evaluating the mobility functions in averaging. The modal loss factor is assumed to be the same for each modes used in averaging. This means that the internal loss factor is the same for each subsystem and is equal to the modal loss factor [11]. The results given are plotted against the modal loss factor in order to show the damping effect at the same time. 3.2 Results and discussion Figure 3 shows the identified coupling loss factor 7/12 for the case (i) {Lo = 1.0777) with different stiffness of /v' 2 . The case of /v '2 = 0 means that the third subsystem Figure 3: 7/12 is negatively affected in three-subsystem model is not present. It can be seen that the coupling loss factor 7^12 is decreased in the low 1158 range of damping while the strength of the coupling between subsystem 2 and 3 is increased. The stronger the coupling, the more ijn is decreased. On the other hand, for the case (ii) where L 2 = l.lm, the different results are shown in figure 4 where the presence of the third subsystem would mainly increase i]i 2 in the low range of damping. The increasing magnitude is also dependent on the strength of coupling between subsystem 2 and 3. The explanation for the different variation trends of ?]i 2 due to the third coupled subsystem between figure 3 and 4 will be given later. Figure 4; 771 ■; is positively affected in three-subsystem model From figure 3 and 4, the effect of damping on the coupling loss factor can also be observed. In the low damping region, increasing damping would increase coupling loss factor. After a certain turnover point, increasing damping would make the coupling loss factor decrease and finally 7712 becomes convergent to a value. This agrees with the conclusions drawn in [10, 11]. It is shown that, even though the length of beam 2 has a slight difference in figure 3 and 4, the converged values are still very close. Thus, the converged value seems not to depend on the variation of coupling strength at A '2 and the structural details, although, with the third subsystem existing in the system, the convergent speed is faster. Therefore, it is reasonable to believe that the converged coupling loss factor at sufficiently high damping only depends on the property of the joint rather than other system properties. This joint- dependent property of coupling loss factor in the high range of damping accords with the assumption in the wave approach. Here, it is convenient to define the convergent region in the figure 3 and 4 as the “joint-dependent zone”. However, before the “joint-dependent zone”, coupling loss factor seems very sensitive to the variation of damping loss factor as well as the strength of coupling between subsystem 2 and 3. It is because in the low damping region the system 1159 modal properties have been playing a major role in determining coupling loss fac¬ tor [10,11]. In general, the coupling loss factor represents the ability of energy transmitted between subsystems. It depends not only on the physical strength of the coupling (e.g., spring stiffness in the examples), but also on the amount of over¬ lap between the modes of two connected subsystems. The higher modal overlap between the modes of two connected subsystem, the more energy is transmitted between the subsystems. As a result, the coupling loss factor will be higher even though the physical strength at the joint is unchanged. If the modes in one subsys¬ tem are distributed exactly the same as those in one another(for instance, two exactly same structures are coupled together), the coupling loss factor would be varied to the maximum, and vice versa. Therefore, this region could be likely defined as “modal-sensitive zone”. In the “modal-sensitive-zone”, the dependence of coupling loss factor on the amount of overlap between the modes of two connected subsystems has been clearly shown in figures 3 and 4. For the case (i) illustrated in figure 3, the length of beam 2 is half of beam 1 . Due to the characteristic of mode distribution in beam structure, the amount of overlap between the modes of subsystem 1 and 2 is more than that in the case (ii) shown in figure 4, where beam 1 is 2 meters and beam 2 is 1.1 meters. Therefore, case (i) has higher coupling loss factor than case (ii) in “modal-sensitive zone”. When the third beam is coupled, the induced variation of coupling loss factor depends on how the amount of overlap between the modes of subsystems 1 and 2 is affected. It can be increased or decreased and thus the coupling loss factor between subsystems 1 and 2 can also be increased or decreased due to the third coupled subsystem. For example, the amount of such overlap in case (i) is decreased after the third subsystem is coupled. As a result, the coupling loss factor, 7712 , becomes decreased. The above discussed variability of coupling loss factor due to the third coupled subsystem has been shown for individual cases. On the other hand, if an ensemble of similar structures are considered, this sensitivity may be reduced (as it is some¬ times positive or negative depending on each special situation). However, such a variability obtained from two typical examples is nevertheless very useful when one individual case is studied in SEA or SEA-like problems. The ignorance of such effect of the other coupled subsystems on the coupling loss factor may become one of the possible error sources causing SEA failure. 4 Conclusions The variation of coupling loss factor due to the third coupled subsystem is stud¬ ied in this paper. The effect of a third coupled subsystem on the coupling loss factor between the first two coupled subsystems depends on how strong the third subsystem is coupled. Roughly, along with the damping in the subsystems, “joint- dependent zone” and “modal-sensitive zone” are defined according to the different variation properties of coupling loss factor. In the “modal-sensitive zone”, the ef- 1160 feet of a third coupled subsystem on the coupling loss factor could be positive or negative. It depends on how the amount of overlap between the modes of two con¬ nected subsystems is affected. This “modal-sensitive” effect may be averaged out for an ensemble of structures, but it is important when SEA is applied to individual cases. In the “joint-dependent zone”, the coupling loss factor is insensitive to the strength of the coupling between the second and third subsystems. Since the two different zones are allocated according to the system damping (which is equivalent to modal overlap factor when the central frequency and modal density are fixed), it shows the importance of reverberance in subsystems when the classical SEA is applied [7,8]. How to take into account the effect of the other coupled subsystems in evaluating coupling loss factor, especially when the system damping is low and when an individual case is considered, definitely needs to be further investigated. References [ 1 ] Richard H. Lyon and Richard G. DeJong. Theory and Application of Statistical Energy Analysis. Butterworth-Heinemann, second edition, 1995. [2] Richard H. Lyon and G. Maidanik. Power flow between linearly coupled os¬ cillators. Journal of the Acoustic Society of America, 34:623-639, 1962. [3] Eric E. Ungar. Statistical energy analysis of vibrating systems. Transactions of the ASME, Journal of Engineering for Industry, pages 626-632, November 1967. [4] F. J. Fahy. Statistical energy analysis. In R. G. White and J. G. Walker, editors. Noise and Vibration, chapter 7, pages 165-186. Chichester, Ellis Horwood, 1982. [5] J. M. Cuschieri and J. C. Sun. Use of statistical energy analysis for rotating machinery, part II; Coupling loss factors between indirectly coupled substruc¬ tures. Journal of Sound and Vibration, 170(2): 191-201, 1994. [6] P. W. Smith. Statistical models of coupled dynamical systems and the transi¬ tion from weak to strong coupling. Journal of the Acoustic Society of America, 65:695-698, 1979. [7] S. Finnveden. Ensemble averaged vibration energy flows in a three-element structure. Journal of Sound and Vibration, 187(3);495-529, 1995. [8] A. J. Keane. A note on modal summations and averaging methods as ap¬ plied to statistical energy analysis (SEA). Journal of Sound and Vibration, 164(1); 143-156, 1993. [9] B. R. Mace. The statistical energy analysis of two continuous one-dimensional subsystems. Journal of Sound and Vibration, 166(3):429-461, 1993. 1161 [ 10] Hongbing Du and Fook Fah Yap. A study of damping effects on coupling loss factor used in statistical energy analysis. In Proceedings of the Fourth Inter¬ national Congress on Sound and Vibration, pages 265-272, St. Petersburge, Russia, June 1996. [11] Fook Fah Yap and J. Woodhouse. Investigation of damping effects on statis¬ tical energy analysis of coupled structures. Journal of Sound and Vibration, 197(3):35I-371, 1996. [12] D. A. Bies and S. Hamid. In situ determination of loss and coupling loss factors by the power injection method. Journal of Sound and Vibration, 70(2): 187-204, 1980. [13] L. Cremer, M. Heckl, et al. Structure-Borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies. Springer-Verlag, second edition, 1987. [14] R. S. Langley. A derivation of the coupling loss factors used in statistical energy analysis. Journal of Sound and Vibration, 141(2):207-219, 1990. [15] J. C. Sun, C. Wang, et ai. Power flow between three series coupled oscillators. Journal of Sound and Vibration, 189(2) :215-229, 1996. 1162 THE EFFECT OF CURVATURE UPON VIBRATIONAL POWER TRANSMISSION IN BEAMS SJ. Walsh(l) and R.G.White( 2 ) ( 1 ) Department of Aeronautical and Automotive Engineering and Transport Studies Loughborough University ( 2 ) Department of Aeronautics and Astronautics University of Southampton ABSTRACT Previous research into structural vibration transmission paths has shown that it is possible to predict vibrational power transmission in simple beam and plate structures. However, in many practical structures transmission paths are composed of more complex curved elements; therefore, there is a need to extend vibrational power transmission analyses to this class of structure. In this paper, expressions are derived which describe the vibrational power transmission due to flexural, extensional and shear types of travelling wave in a curved beam which has a constant radius of curvature. By assuming sinusoidal wave motion, expressions are developed which relate the time-averaged power transmission to the travelling wave amplitudes. The results of numerical studies are presented which show the effect upon power transmission along a curved beam of: (i) the degree of curvature; and (ii) various simplifying assumptions made about the beam deformation. 1 . INTRODUCTION Previous research into structural transmission paths has shown it is possible to predict vibrational power transmission in simple beam and plate structures. More recently, transmission through pipes with bends, branches and discontinuities has been studied, which has led to useful design rules concerning the position and size of pipe supports for minimum power transmission[l]. However, in many practical structures transmission paths are composed of more complex curved elements. Therefore, there is a need to extend power transmission analyses to this class of structure. 1163 Wave motion in a curved beam with a constant radius of curvature has been considered by Love [2] who assumed that the centre-line remains unextended during flexural motion, whilst flexural behaviour is ignored when considering extensional motion. Using these assumptions the vibrational behaviour of complete or incomplete rings has been considered by many researchers who are interested in the low frequency behaviour of arches and reinforcing rings. In reference [2] Love also presented equations for thin shells which include the effects of extension of the mid-surface during bending motion. Soedel [3] reduced these equations and made them applicable to a curved beam of constant radius of curvature. In an alternative approach Graff [4] derived these equations from first principles and also constructed frequency verses wavenumber and wavespeed versus wavenumber graphs. Philipson [5] derived a set of equations of motion which included extension of the central line in the flexural wave motion, and also rotary inertia effects. In a development analogous to that of Timoshenko for straight beams, Morley [6] introduced a correction for radial shear when considering the vibration of curved beams. Graff later presented frequency versus wave number and wave speed versus wave number data for wave motion in a curved beam, when higher order effects are included [7]. In this paper, expressions for vibrational power transmission in a curved beam are derived from first principles. In the next section two sets of governing equations for wave motion in a curved beam are presented both of which include coupled extensional-flexural motion. The first set is based upon a reduction of Love's thin shell equations mentioned above. The second set is based upon a reduction of Fliigge’s thin shell equations [8]. In section three, the expressions for stresses and displacements presented in section two are used to derive formulae for vibrational power transmission in terms of centre-line displacements. By assuming sinusoidal wave motion, expressions are developed which relate the time-averaged power transmission to the extensional and flexural travelling wave amplitudes. In section four, corrections for rotary inertia and shear deformation are introduced. The results of numerical studies of these expressions are presented which show the effect upon wave motion and power transmission of (i) the degree of curvature, and (ii) the various simplifying assumptions made about the beam deformation. 1164 2. WAVE MOTION IN CURVED BEAMS In this section the governing relations between displacements, strains, stresses and force resultants in a curved beam are presented. The centre-line of the beam lies in a plane and forms a constant radius of curvature. The cross-section of the beam is uniform and symmetrical about the plane and it is assumed that there is no motion perpendicular to the plane. It is also assumed that the beam material is linearly elastic, homogeneous, isotropic and continuous. Consider a portion of the curved beam, as shown in Figure 1. The circumferential coordinate measured around the centre-line is s, while the outward pointing normal coordinate from the centre¬ line is z, and the general radial coordinate is r. A complete list of notation is given in the appendix. For small displacements of thin beams the assumptions, known as "Love's first approximation" in classical shell theory, can be made [8]. This imposes the following linear relationships between the tangential and radial displacements of a material point and components of displacement at the undeformed centre-line: U (r, s, t) = u (R, s, t) + z (|) (s, t) (1) W (r, s, t) = w (R, s, t) (2) where u and w are the components of displacement at the centre¬ line in the tangential and radial directions, respectively, (j) is the rotation of the normal to the centre-line during deformation: (|> 3w / angle of'X {rotational displacement^ Vcurvature/ \ of straight beam ) (3) and W is independent of z and is completely defined by the centre¬ line component w. Circumferential strain consists of both an extensional strain and bending strain component. Expressions for these are listed in table 1. The strain-displacement expressions of the Love and Flugge based equations are identical. However, in the total circumferential strain of the Love based theory, the term in the denominator has been neglected with respect to unity. Assuming the material to be linearly elastic, the circumferential stress-strain relationship is given by Hooke's Law, whilst the shear strain, Ysr/ and shear stress, Osr/ are assumed to be zero. Assuming the material 1165 to be homogeneous and isotropic, the material properties E, G and v can be treated as constants. Thus, by integrating the stresses over the beam thickness, force and moment resultants can be obtained, which are listed in table 2. The adopted sign convention is shown in Figure 2. Equations of motion for a curved beam are presented in [4]. These equations are derived in terms of the radian parameter 0. By applying the substitution, s - RQ, the equations of motion can be expressed in terms of the circumferential length, s. These equations are listed in [9] along with the Fliigge based equations of motion which have been obtained by a reduction of the equations of motion for a circular cylindrical shell presented in [8]. An harmonic solution of the equations of motion can be obtained by assuming that extensional and flexural sinusoidal waves propagate in the circumferential direction. The harmonic form of the equations of motion are also listed in [9]. 3. VIBRATIONAL POWER TRANSMISSION IN CURVED BEAMS In this section the expressions for displacements and stresses presented in section two are used to derive the structural intensity and power transmission due to flexural and extensional travelling waves in a curved beam. The structural intensity expressions are formulated in terms of displacements at the centre-line. By assuming sinusoidal wave motion, expressions are developed which relate the time-averaged power transmission to the flexural and extensional travelling wave amplitudes. Structural Intensity in the circumferential direction of a curved beam is given by [10]: Is = au at f intensity due to I circumferential stress aw at f intensity due to radial shear stress ( 4 ) By integrating across the beam thickness power transmission per unit length in the circumferential direction is obtained: h/2 Ps= J Isdz (5) -h/2 1166 Substituting the Love based circumferential stress-strain relation and strain-displacement expression into equation (5) the power transmission due to circumferential stress is obtained. (A full derivation is given in [9].) By analogy to power transmission in a straight beam [1] this can be expressed in terms of an extensional component, Pe, and a bending moment component, Pbm* Although the transverse shear stress Ogr is negligible under Love's first approximation, the power transmission due to transverse shear stress can be evaluated from the non-vanishing shear force, Q, because the radial displacement W does not vary across the beam thickness. Again, by analogy to power transmission in a straight beam [1] this is expressed as a shear force component. Thus, the total power transmission in the circumferential direction is given by the sum of the extensional, bending moment and shear force components. These equations are listed in table 3 along with Fliigge based power transmission equations which are also listed in table 3. Substituting harmonic wave expressions into the Love and Fliigge based power transmission equations gives expressions for the power transmission in the circumferential direction in terms of travelling wave amplitudes A and B. For sinusoidal wave motion it is useful to develop time-averaged power transmission defined by [1]: T/2 <Ps> -- f t“T J Ps (s,t) dt - T/2 ( 6 ) where T is the period of the signal. Time averaged Love and Fliigge based power transmission equations are given in table 4. 4. THE EFFECT OF ROTARY INERTIA AND SHEAR DEFORMATION It is known that shear deformation and rotary inertia effects become significant for straight beams as the wave length approaches the same size as the thickness of the beam, and for cylindrical shells as the shell radius decreases [8]. Thus, the objective in this section is to establish more complete equations for power transmission in a curved beam and to show under what conditions these specialise to the simple bending equations presented in section three. Rotary inertia effects are included by considering each element of the beam to have rotary inertia in addition to translational inertia. Equations of motion for a curved beam which include the 1167 effect of rotary inertia are presented in [7]. These equations are listed in [9] in terms of the circumferential distance parameter, s. Equations for vibrational power transmission can be derived in the same marmer as described in section three. These equations are listed in tables 3 and 4 where it can be seen that the extensional and bending moment components when including rotary inertia effects are identical to the corresponding Fliigge based expressions. However, the shear force component now contains an additional rotary inertia term. If shear deformation is included then Kirchoff s hypothesis is no longer valid, and the rotation of the normal to the centre-line during bending, (|), is no longer defined by equation (3) but is now another independent variable related to the shear angle, y. However, unlike simple bending theory, where the transverse shear strain, Ysr^ is negligible, the transverse shear strain is now related to the shear angle, y which is expressed in terms of displacements u, w and (j). The circumferential force, bending moment, and shear force obtained from [8] are given in table 2. A set of equations of motion for a curved beam which includes the effect of shear deformation is presented in [9]. Power transmission equations in the circumferential direction can be obtained in a manner analogous to that used for Love and Fliigge based theories. As before, the power transmission due to circumferential stress can be identified as consisting of extensional and bending moment components. The contribution to the power transmission from the transverse shear stress is obtained from the product of the shear force resultant and the radial velocity which gives the shear force component of power transmission. 5. NUMERICAL STUDY For a given real wavenumber, k, the harmonic equations of motion were solved to find the corresponding circular frequency, co, and complex wave amplitude ratio. The simulated beam was chosen to have the physical dimensions and material properties of typical mild steel beams used for laboratory experiments. Four different radii of curvature were investigated, which were represented in terms of the non-dimensional thickness to radius of curvature ratio, h/R. These ratios were ^/lO/100/1000 and ^/lOOOO- Using the Love equations of motion. Figure 3 shows the relationship between wave number and frequency for a beam with a 1168 thickness to radius of curvature ratio of ^/lo- The frequency range is represented in terms of the non-dimensional frequency parameter Q = ®^/co, where Cq is the phase velocity of extensional waves in a straight bar and the wave number range is represented in terms of the non-dimensional wave number, kR. It can be seen that two types of elastic wave exist: one involving predominantly flexural motion; the other predominantly extensional motion. However, for wave numbers less than kR = 1, the predominantly flexural wave exhibits greater extensional than flexural motion. Solution of the shear deformation equations of motion for a curved beam shows that three types of elastic wave exist. These are the predominantly flexural and predominantly extensional waves of simple bending theory and additionally a predominantly rotational wave related to the shear angle. The relationship between wave number and frequency for these three wave types is shown in Figure 4. A numerical investigation of the power transmission equations was undertaken using simulated beams with the same dimensions and material properties as those used in the previous study of wave motion. Figure 5 shows the relationship between transmitted power ratio and frequency. For the predominantly flexural wave the time-averaged transmitted power ratio is calculated by dividing the time-averaged power transmitted along a curved beam by a predominantly flexural wave by the time- averaged power transmitted by a pure flexural wave travelling in a straight Euler-Bernoulli beam. i.e. the ratio (<Pe>t + <Fbm>t + <Psf>t)/EIcokf3Af. For the predominantly extensional wave the transmitted power ratio is calculated by dividing the time-averaged power transmitted along a curved beam by a predominantly extensional wave by the time-averaged power transmitted by a pure extensional wave in a straight rod. i.e. the ratio (<Pe>t + <Pbn>t + <Psf >t)/EScokex P^ex 6. SUMMARY AND CONCLUSIONS In this paper, starting from first principles, expressions for vibrational power transmission in a curved beam have been derived using four different theories. Love’s generalised shell equations include extension of the centre-line during bending motion were the first set of equations considered. Fiiigge's equations also include centre-line extensions and were the second set of equations used. Corrections for rotary inertia and shear deformation produced the third and fourth sets of governing 1169 equations, respectively. By letting the radius of curvature, R, tend to infinity these equations reduce to the corresponding straight beam expressions presented in [1]. Using the governing equations for each theory, expressions were then developed which related time-averaged power transmission to the amplitudes of the extensional, flexural and rotational displacements. For each theory the effects of curvature upon the resulting wave motion and power transmission were then investigated using beams with different degrees of curvature. From the results of this study it can be seen that vibrational power transmission in curved beams can be classified into three different frequency regions: (i) below the ring frequency, Q = 1, curvature effects are important; (ii) above the ring frequency but below the shear wave cut-on frequency, = 1 the curved beam behaves essentially as a straight beam; (iii) above the shear wave cut-on frequency, higher order effects are important. ACKNOWLEDGEMENT The analytical work presented in this paper was carried out while both authors were at the Institute of Sound and Vibration Research, University of Southampton. The financial support of the Marine Technology Directorate Limited and the Science and Engineering Research Council is gratefully acknowledged. REFERENCES 1. J.L. HORNER 1990 PhD thesis, University of Southampton Vibrational power transmission through beam like structures. 2. A.E.H. LOVE 1940 Dover, Nezv-York. A treatise on the mathematical theory of elasticity. 3. W. SOEDEL 1985 Dekker, New York. Vibrations of shells and plates 4. K.F. GRAFF 1975 Clarenden Press, Oxford. Wave motion in elastic solids 5. L.L. PHILIPSON 1956 Journal of Applied Mechanics 23, 364- 366. On the role of extension in the flexural vibrations of rings. 6. L.S.D. MORLEY 1961 Quarterly Journal of Mechanics and Applied Mathematics 14, (2), 155-172. Elastic waves in a naturally curved rod. 1170 7. K.F. GRAFF 1970 IEEE Transactions on Sonics and Ultrasonics, SU-17 (1), 1-6. Elastic wave propagation in a curved sonic transmission line. 8. A.W. LEISSA 1977 NASA SP-288, Washington DC Vibrations of shells. 9. S.J. WALSH 1996 PhD thesis, University of Southampton. Vibrational power transmission in curved and stiffened structures. 10. A.J. ROMANO, P.B. ABRAHAM, E.G. WILLIAMS 1990 Journal of the Acoustical Society of America 87. A Poynting vector formulation for thin shells and plates, its application to structural intensity analysis and source localization. Part I: Theory. APPENDIX: NOTATION A flexural wave amplitude Af amplitude of a purely flexural wave B extensional wave amplitude Bex amplitude of a purely extensional wave C rotation wave amplitude E Young’s modulus G shear modulus I second moment of area of cross-section of beam Ig structural intensity in circumferential direction K radius of gyration M bending moment on cross-section of beam N circumferential force on cross-section of beam P transmitted power Q shear force on cross-section of beam R radius of curvature S cross-sectional area of beam T period of wave LF displacement in circumferential direction W displacement in radial direction Co wavespeed of extensional waves in a straight bar Cs wavespeed of shear waves in a straight bar ds length of elemental slice of curved beam eg total circumferential strain h thickness of beam k wavenumber kex wavenumber of a purely extensional wave kf wavenumber of a purely flexural wave r coordinate in radial direction 1171 s coordinate in circumferential direction u displacement at centre-line in circumferential direction w displacement at centre-line in radial direction z coordinate of outward pointing normal Q. non-dimensional frequency Ps bending strain Y shear angle Ysr transverse shear strain Ej. radial strain £g circumferential strain 0A phase angle of flexural wave 0C phase angle of rotational wave K Timoshenko shear coefficient ?iex wave length of extensional waves in a straight bar V Poisson's ratio Oj. radial stress Gg circumferential stress 0 SJ transverse shear stress d change in slope of normal to centre-line CO radian frequency W(r,s,t) 1172 Figure 2: Sign convention and force resultants on an elemental slice of curved beam Figure 3: Wave number v. frequency relationship for a curved beam predicted using Love theory Transmitted power ratio (Curved beam)/(Slraight beam) ^ Non-dimensional wavenumber Table 1 : Displacement, strain-displacement and stress-strain equations for a curved beam. 1175 Table 2: Force resultants for a curved beam. 1176 Table 3: Power transmission for a curved beam. 1177 Table 4 : Time-averaged power transmission by a single harmonic wave 1178 A Parameter-based Statistical Energy Method for Mid-frequency Vibration Transmission Analysis Sungbae Choi, Graduate Student Research Assistant Matthew P. Castanier, Assistant Research Scientist Christophe Pierre, Associate Professor Department of Mechanical Engineering and Applied Mechanics The University of Michigan Ann Arbor, MI 48109-2125, USA Abstract Vibration transmission between two multi-mode substructures con¬ nected by a spring is investigated. A classical Statistical Energy Analysis (SEA) approach is reviewed, and it is seen that some typ¬ ical assumptions which are valid at high frequencies lose accuracy in the mid-frequency range. One assumption considered here is that of an identical probability density function (pdf) for each reso¬ nant frequency. This study proposes a Parameter-based Statistical Energy Method (PSEM) which considers individual modal informa¬ tion. The results of PSEM have good agreement with those of a Monte Carlo technique for an example system. Nomenclature E[ ] expected value power transmitted between substructure i and k IIfc(u;) . total power transmitted to substructure k Pi power input to substructure i Vi, coupling loss factor (CLF) fh modal driving force for mode j of substructure 1 UJ frequency [rad/sec] l.l.-.lj.Ol subscripts for decoupled Bar 1 2,2r,2„02 subscripts for decoupled Bar 2 El, ^2 blocked energy Eoi, Eo2 Young’s modulus Pl^P2 density mi,m2 mass per unit length Mu M 2 total mass Cl,C2 viscous damping ratio Ai, A 2 cross-sectional area 0 rH 0 nominal length 1179 Ti,T2 disordered length £1,^2 ratio of disorder to nominal length k coupling stiffness coupling ratio Xi,X2 position coordinate ai,a2 point-coupling connection position Wi, W2 deflection modal amplitude mode shape function NuN2 number of resonant frequencies resonant frequencies lower limit of resonant frequencies upper limit of resonant frequencies (Tl, (72 standard deviation of disorder 1 Introduction Vibration transmission analysis between connected substructures in the mid-frequency range is often a daunting prospect. Since the analysis at high frequencies requires greater model discretization, the size and com¬ putational cost of a full structure model (e.g., a Finite Element model) can become prohibitive. Also, as the wavelengths approach the scale of the structural variations, uncertainties (tolerances, defects, etc.) can sig¬ nificantly affect the dynamics of the structure. Starting at what may be called the mid-frequency range, deterministic models fail to predict the response of a representative structure with uncertainties. Therefore, in the mid-frequency range, a statistical analysis of vibra¬ tion transmission may be more appropriate. This approach is taken in the procedure known as Statistical Energy Analysis (SEA) [1]. In SEA, a structure is divided into coupled substructures. It is assumed that each substructure exhibits strong modal overlap which makes it difficult to dis¬ tinguish individual resonances. Therefore, the resonant frequencies are treated as random variables, each with an identical, uniform probability density function (pdf) in the frequency range of interest. This assumption greatly simplifies the evaluation of the expected value of transmitted vi¬ bration energy. A simple linear relation of vibration transmission between each pair of substructures is retrieved. The power transmitted is propor¬ tional to the difference in the average modal energies of the substructures. This relation is analogous to Fourier’s law of heat transfer [1-4]. In the low- to mid-frequency region, the modal responses are not strongly overlapped. In this case, two typical SEA assumptions are less accurate: an identical pdf for all resonant frequencies, and identical (ensemble-averaged) 1180 values of the associated mode shape functions at connection positions. In this paper, these two assumptions are relaxed. A distinct uniform pdf is applied for each resonant frequency, and a piecewise evaluation of the transmitted power is performed. This is called a Parameter-based Sta¬ tistical Energy Method (PSEM) because it considers the statistical char¬ acteristics of individual system parameters. This solution can accurately capture peaks of transmitted power while maintaining the SEA advantage of efficiency. This paper is organized as follows. In section 2, we briefly review SEA along with the associated assumptions and limitations. In section 3, the power transmitted between two spring-coupled multi-mode substructures is investigated by applying several SEA assumptions. A Monte Carlo solu¬ tion is used for comparison. In section 4, the PSEM approach is presented and the results are shown. Finally, section 5 draws conclusions from this study. 2 Overview of SEA In Statistical Energy Analysis, the primary variable is the time-averaged total energy of each substructure. This is called the blocked energy, where blocked means an assumed coupling condition. The assumed coupling con¬ dition may be the actual coupling, a clamped condition at the substructure junctions, or a decoupled condition [2, 5]. In order to predict the average power transmitted between two directly- coupled substructures, a few simplifying assumptions are applied. Some of the essential SEA assumptions are summarized by Hodges and Woodhouse in Ref. [3]; • Modal incoherence: the responses of two different modal coordinates are uncorrelated over a long time interval • Equipartition of modal energy: all modes within the system have the same kinetic energy The above conditions make it possible to treat all modal responses as sta¬ tistically identical. The first assumption implies a broad band, distributed driving force (often called ”rain on the roof’) which leads to uncorrelated modal driving forces. The second assumption implies that the substruc¬ tures have strong modal overlap, or that the parameter uncertainties are sufficiently large that the modes are equally excited in an ensemble average sense. Thus, the resonant frequencies are treated as random variables with identical, uniform probability density functions (pdfs) for the frequency range of interest. The SEA relation for the expected value of power transmitted from 1181 substructure i to substructure k, may be expressed as E[nijfc(cj)] = 0)7},^Ni{^ (1) where uj is the frequency, 77 ^^ is the coupling loss factor, Ei is the blocked energy of substructure z, and Ni is the number of participating modes of substructure i for the frequency range of interest. The power dissipated by substructure i is expressed as = u:riiEi (2) where r]i is the damping loss factor. Using Eqs.(l) and (2), the equation of power balance for substructure i at steady state [1, 2, 5, 6] is N N Pi = E[Ilij] + UJTJiEi = UJ ( 3 ) i=i j¥i where Pi is the power input to substructure i from external sources. Note that the first term on the right-hand side is the power transmitted through direct coupling between substructures. 3 Vibration Transmission in a Two-Bar System The longitudinal vibration of the structure shown in Fig.l is considered in this study. The structure consists of two uniform bars with viscous damping which are coupled by a linear spring of stiffness k. The spring is connected at intermediate points on the bars, Xi = ai and X 2 = 0 , 2 . Bar i has nominal length Loi. A parameter uncertainty may be introduced by allowing the length to vary by a small random factor e^, which is referred to as disorder. The length of a disordered bar is Li = Zrox(l + £:)• The ratio of the connection position to the length, aifLi^ is held constant. Bar 1 is excited by a distributed force Fi{xi,t). 3.1 Nominal transmitted power The power transmitted from Bar 1 to Bar 2 for the nominal system (no disorder) is briefly presented here. A more detailed derivation is shown in the Appendix (see also Refs. [7] and [8]). The equations of motion are Wi(a:i,t) -f ^Ci(wi(a;i,f)) = Fi(a:i,t)-|- ^[W2(a2, t)-wi(ai, - ci) (jBo2A2-§^ + m2|^^W2(x2,t) + ^C'2(w2(a;2,t)) = k[wi{ai,t)-W2{a2,t)]6{x2 - ^ 2 ) 1182 a) Bar 2 Fig. 1: Two-bar system where ^ is a Dirac delta function, and (for Bar i) Eoi is Young’s modulus, Ai is the cross-sectional area, rm is the mass per unit length, ’Ci is the viscous damping operator, and Wt(a:i,t) is the deflection. The deflections of the two bars can be expressed by a summation of modes: Wi{xi,t) = W2(x2,t) = E 1^2r(i)^2r(^2) (5) t=0 r=0 where and W 2 ^{t) are modal amplitudes, and and ^ 2 r(^ 2 ) are mode shape functions of the decoupled bars. These mode shape func¬ tions are normalized so that each modal mass is equal to the total mass of the bar, Mi. Applying modal analysis and taking a Fourier transform, the following equations are obtained: Mi(f)ijWij = 4- fc^ij(ai)[E kF2r^2r(<^2) - E _ (6) M2<f>2,W2, = ^^^2,(<22)[E VFi,^i,(ai) - E W2.^2M2)] i=0 r=0 [ul. - -f • 2Cia;i .a;)(2 - sgn(j)) y/^ • 2 C 2 <^ 2.^)(2 - sgn(s)) 1 for ^ > 0 0 for 2 = 0 where an over-bar (") denotes a Fourier transform, and are resonant frequencies, (i and (2 are damping ratios, and is a modal driving force. Mode 0 is a rigid body mode, which is why the sgn(i) term is present. Note that the damping ratio of each bar is assumed to be the same for all modes. Next, the modal driving forces are assumed to be incoherent, and each spectral density function is assumed to be constant (white noise) over a finite range of frequency [2]. After some algebra, the power transmitted <^2^ = sgn(0 = 1183 (7) from Bar 1 to Bar 2, 1112 ( 0 ;), is found as: 2o;^PC2«^PlPl ^ ^2r(^2)0^2r M1W2IAI2 ^ |(^ 2.|2 -^1 -^2 <?^ 2 r where 5pjpi is the same uniform spectral density function for each modal driving force on Bar 1. 1112(0;) = A = 3.2 Monte Carlo Energy Method (MCEM) The disordered case is now considered, where each bar has a random length. The ensemble-averaged transmitted power for a population of disordered two-bar systems is found by tahing the expected value of Eq. (7): ^[ni2] ( 8 ) where Ni is the number of modes taken for Bar i (this is an arbitrary set of modes that have been aliased to the numbers 1,2,- • ’,Ni). Since a truncated set of modes is used, Eq. (8) is an approximation. The random variables in Eq. (8) are the resonant frequencies of the bars (which are present in the terms ^i;, ^ 2 r? s,nd A). Equation (8) may be solved numerically using a Monte Carlo method: the random variables are assigned with a pseudo-random number generator for each realization of a disordered system, and the transmitted power is averaged for many realizations. This is called a Monte Carlo Energy Method (MCEM) here. It may be used as a benchmark for comparing the accuracy of other approximate methods. Note that the resonant frequencies of a bar may be found directly from the disordered length. Therefore, for the MCEM results in this study, the actual number of random variables in Eq. (8) is taken to be one for each decoupled bar. That is, the two random lengths are assigned for each real¬ ization, and then the natural frequencies are found for each bar in order to calculate the transmitted power. If such a relation were not known, each resonant frequency could be treated as an independent random variable. 3.3 SEA-equivalent Transmitted Power An SEA approximation of the transmitted power may be obtained by ap¬ plying several typical SEA assumptions to Eq. (8). (Since Eq. (1) is not used directly, this might be called an SEA-equivalent transmitted power.) These assumptions were summarized in Ref. [8]: the coupling between 1184 substructures is weak, the modal responses are uncorrelated, the expected value of the square of mode shape functions at connection positions is unity, and the pdfs of the resonant frequencies are uniform and identical. The assumption of weak coupling means that the value of jAj in Eq. ( 8 ) is approximately one. Applying the second and third assumptions then yields E[ai2] M?M2 (9) Since the pdf of each resonant frequency is assumed to be uniform, the expected values in Eq. (9) are E[ E[ l/4u)" 1 . 2 cos| -f 2 a;a;i. • cosf + ^ ul. — 2u}ujii • cos| • • sin I sin% ian~ U) 2 _ ( 10 ) l/4a;"C2Vr^, - 1) - 1 - tan ——--— W 2 r =‘*' 2 ru — ^2r 4w^C2\/i - Cl (11) W2r=‘*'2rj where a = cos~^(l - 2 Cj), subject to the restrictions (1 - 2(1)^ < 1 and (1 _ 2 ( 2)2 ^ I Finally, since the pdfs of the resonant frequencies are taken to be identical, the frequency limits do not depend on the individual modes (a;i. = uj 2 r, - a^nd = u; 2 ,„ = oju)- Therefore, each sum in Eq. (9) simplifies to the product of the expected value and the number of modes in the frequency range of interest: E[n^2] 2C2A:ViViiV25,^p^ r.r 1 1 Pr ^2. ^ WM2 Vi.r ( 12 ) Equation (12) is the SEA approximation used in this study. 3.4 Example The three formulations of the transmitted power presented thus far the nominal transmitted power in Eq. (7), the MCEM transmitted power in Eq. ( 8 ), and the SEA-equivalent transmitted power in Eq. (12) — are now compared for a two-bar system with the parameters shown in Table 1. For the MCEM results, the disorder (ei and 62 ) was taken to be uniformly distributed with mean zero and standard deviation ai = 0-2 = 10%. As a measure of the coupling strength, the coupling ratio, Ri, is defined cLS the ratio of coupling stiffness to the equivalent stiffness of a bar at the 1185 Table 1: Material properties and dimensions of two bars MIIM 2 21.53/21,53 [Kg] R 01 IL 02 10.58/8.817 [m] E 01 IR 02 200 XIOV2OO xlO^ [N/m'1 ai/a2 2.116/7.053 N Pllp2 7,800/7,800 [Kg/m'*l ^pipi 1 [N^l C1/C2 0.005/0.005 k 4.868 xlO^ [N/m; Fig. 2: Comparison of the nominal transmitted power, the MCEM results (20,000 realizations with (j^ = <72- 10%), and the SEA approximation. fundamental resonant frequency, Ri = weak cou¬ pling is considered here such that Ri = 0.01. The nominal transmitted power, the MCEM results, and the SEA ap¬ proximation are shown in Fig.2. The transmitted power calculated for the nominal system exhibits distinct resonances. This is due to the low modal overlap of the bars in this frequency range. The MCEM results show distinct peaks for uj < 15,000 rad/s, but they become smooth as the frequency increases. The SEA approximation does not capture indi¬ vidual resonances. However, at the higher frequencies where the disorder effects are stronger, the SEA approximation agrees well with the MCEM results. The frequency range between where the MCEM results are close to the nominal results and where they are close to the SEA results (ap¬ proximately 2,500 ~ 15,000 rad/s for this case) is considered to be the mid-frequency range here. This range will vary depending on the system 1186 parameters and the disorder strength. In the next section, an efficient ap¬ proximation of the transmitted power is presented which compares well with MCEM in the mid-frequency range. 4 Parameter-based Statistical Energy Method (PSEM) The SEA approximation presented in the previous section does not capture the resonances in the transmitted power because of two assumptions: the resonant frequencies all have the same uniform pdf, and the values of the square of mode shape functions at the connection positions are taken to be the ensemble-averaged value. Keane proposed an alternate pdf of resonant frequencies in order to apply SEA to the case of two coupled nearly periodic structures [9]. This pdf is shown in Fig.3(a). It accounts for the fact that Fig. 3: (a) The pdf of the natural frequencies and the resultant transmitted power from Ref. [9]. (b) The pdfs of three natural frequencies, and a schematic representation of the piecewise evaluation of transmitted power for PSEM. The individual modal contributions are extrapolated (-) and summed to calculate the total transmitted power (—). the natural frequencies of a nearly periodic structure tend to be grouped in several distinct frequency bands. Thus the pdf has a large constant value for those frequency bands, and a small constant value elsewhere. The SEA approximation of transmitted power is then modified by simply adding a positive value or negative value on a logarithmic scale, as demonstrated in Fig, 3(a). This solution thus captures some of the resonant behavior of the transmitted power. 1187 Here, a more general approach is taken for approximating the power transmitted between two substructures in a frequency range in which they have low or intermediate modal overlap. Each resonant frequency is as¬ signed a uniform pdf. However, the frequency range of each pdf is diiferent; it corresponds to the range in which that resonant frequency is most likely to be found. (The concept of using “confidence bands” as one-dimensional pdfs was suggested but not pursued in Ref. [8].) An example is shown in Fig. 3(b) for three resonant frequencies. Furthermore, it is assumed that the values of the square of the mode shape functions at the connection positions are known. Thus, applying only the first two SEA assumptions along with those noted above, Eq. (8) becomes: N2 i:kme This is called a Parameter-ba.sed Statistical Energy Method (PSEM) be¬ cause it employs information for individual modal parameters. Since each modal pdf is uniform, Eqs. (10) and (11) still hold for the expected values in Eq. (13). However, unlike the SEA approximation, each expected value is different, because the corresponding frequency bounds are unique. Furthermore, note that the pdfs do not cover the entire frequency range of interest. The results for each mode are therefore extrapolated outside the frequency range of that modal pdf before the individual modal contributions are summed. This is shown schematically in Fig. 3(b). PSEM is therefore a piecewise evaluation of the expected value of transmitted power. The PSEM approximation is now applied to the two-bar system of Table 1, with the standard deviation of disorder cti = 0-2 = 10%. The pdfs of the resonant frequencies of Bar 1 and Bar 2 aze shown in Fig. 4(a) and (b), respectively. For this system, the bounds for each resonant frequency may be found directly from the variation of the uncertain parameter. It can be seen that the spread of each natural frequency pdf due to disorder increases with increasing frequency. The MCEM, PSEM, and SEA approximations for the transmitted power are shown in Fig. 4(c). There were 20,000 realizations taken for the MCEM results at each sampled frequency. This took about 10 hours of computa¬ tion time. In contrast, the PSEM results only required 3 seconds of compu¬ tation time, and the SEA results only required about 1 second. Note that the PSEM results show excellent agreement with the much more expensive MCEM results. The difference at very low frequencies comes from the fact that for the PSEM approximation, the value of the term |A| was assumed to be one due to weak coupling. This assumption breaks down as the frequency approaches zero. However, the match between the MCEM and 1188 Fig. 4: (a) Natural frequency pdfs for Bar 1. (b) Natural frequency pdfs for Bar 2. (c) Transmitted powers obtained by MCEM, PSEM, and SEA for c7i = <J2 = 10%. PSEM results in the mid-frequency range is excellent. Again, it is noted that the SEA results converge to those of MCEM (and PSEM) as the fre¬ quency increases. Now it can be seen that the assumption of identical pdfs for all modes becomes better with increasing frequency. Next, the example system is considered with smaller disorder, Ci = (72 = 1%. Fig. 5 shows the results for this case for what might be called the mid-frequency range. Note that even though this is a higher frequency range than that considered for the previous ca.se, the pdfs of the resonant frequencies shown in Fig. 5(a) and (b) are not as strongly overlapped. Thus, several peaks are seen in the transmitted power in Fig. 5(c). Again, the PSEM approximation agrees well with the MCEM results, although there is more discrepancy for this case. The SEA approximation follows the global trend, but does not capture the resonances or anti-resonances. The SEA results drop off at the edges because only modes within this frequency range are considered to contribute to the transmitted power. In addition to PSEM, another piecewise evaluation of the transmitted power is considered here. For this approximation, wherever the individual mode pdfs overlap, they are superposed to form a pdf for all the modes in that “section” of the frequency range. This superposition is demonstrated in Fig. 6. Also, if the number of modes in a section is above a certain cutoff number, Nc, then it is assumed that their mode shape function values at the connection positions are unknown, so that the ensemble-averaged value must be used. This is called a multiple mode approximation. The purpose 1189 Fig. 5: (a) Natural frequency pdfs for Bar 1. (b) Natural frequency pdfs for Bar 2. (c) Transmitted power obtained by MCEM, PSEM,-and SEA for cTi = 0-2 = 1 %. Fig. 6: Resonant frequency pdfs for PSEM and for the multiple mode approximation. of formulating this approach is to investigate what happens as information about the individual modes is lost. The multiple mode approximation is applied to the example system with cTi = <72 = 10% in Fig.7. For A/c = 2, this approximation has good agreement with MCEM. The match is especially good for u) < 10,000. Above this frequency, the number of overlapped resonant frequencies in each pdf section is greater than Nc, and the loss of mode shape information affects the results slightly. For Nc = 0, the values of the mode shape functions are taken to be one for the entire frequency range, just as in the SEA approximation. As can be seen in Fig. 7, the piecewise construction of the pdf roughly captures the frequency ranges of the resonances and anti-resonances. However, the mode shape effect is more pronounced in 1190 -B.S - MCEM -e.s — Multiple mode approximation rvt= 2 . Multiple mode approximation Nfc= O Fig. 7: Transmitted power obtained from MCEM and the multiple mode approximation for Ui = <72 = 10% the mid-frequency range. The peak values are now similar to the SEA approximation. 5 Conclusions In this study, the power transmitted between two multi-mode substruc¬ tures coupled by a spring was considered. A Monte Carlo Energy Method (MCEM) was used to calculate the ensemble average of the transmitted power for the system with parameter uncertainties. A classical Statistical Energy Analysis (SEA) approximation matched the Monte Carlo results in the high-frequency range, but did not capture the resonant behavior of the transmitted power in the mid-frequency range where the substructures have weak modal overlap. A Parameter-based Statistical Energy Method (PSEM) was presented which uses a distinct pdf for each natural frequency as well as some indi¬ vidual mode shape information. A piecewise evaluation of the transmitted power was performed, and then the modal contributions were extrapolated and superposed. The PSEM approximation compared very well with the much more expensive Monte Carlo results, including in the mid-frequency 1191 range. References 1 R. H. Lyon. Statistical Energy Analysis of Dynamical Systems: Theory and Applications. M.I.T. Press, 1st edition, 1975. 2 R. H. Lyon. Theory and Application of Statistical Energy Analysis. Butterworth-Heinemann, 2nd edition, 1995. 3 C. H. Hodges and J. Woodhouse. Theories of noise and vibration trans¬ mission in complex structures. Rep.Prog.Physics, 49:107-170, 1986. 4 J. Woodhouse. An approach to the theoretical background-of statis¬ tical energy analysis applied to structural vibration. Journal of the Acoustical Society of America, 69(6):1695-1709, 1981. 5 M. P. Norton. Fundamentals of Noise and Vibration Analysis for En¬ gineers. Cambridge University Press, 1st edition, 1989. 6 N. Lalor. Statistical energy analysis and its use as an nvh analysis tool. Sound and Vibration, 30(l):16-20, 1996. 7 Huw G. Davies. Power flow between two coupled beams. Journal of the Acoustical Society of America, 51(1):393-401, 1972. 8 A. J. Keane and W. G. Price. Statistical energy analysis of strongly coupled systems. Journal of Sound and Vibration, 117(2):363-386, 1987. 9 A. J. Keane. Statistical Energy Analysis of Engineering Structures (Ph.D Dissertation). Brunei University (England), 1988. Appendix In this appendix, the nominal transmitted power in Eq. (7) for the mono- coupled two-bar system is derived. The procedure follows that of Refs [7, 8 ]. Plugging Eq. (5) into Eq. (4), F^ixut) + fcf E - Oi) (A.1) t=0 1192 k £ Wi,(<)$i,(a,) - £ lV2,(«)«2,(a2) 5 (x 2 - aj) Multiplying Eq. (A.l) by and integrating with respect to xi for [0, Li] yields MiK + = A, + ff; W2,(4)W2 Xo2) - E Li 2CijUijMi = y C'i(«'i,.(a:i))Wij(xi)cia:i , fi, J Fi{xi,t)^i.{xi)dx], 0 0 and wij is the ;th resonant frequency of decoupled Bar 1. The damping ratio in Eq. (A.3) is now assumed to be the same (Ci) for all modes, since the differences in the ratio are usually small and this simplifies the equation. Taking the Fourier transform of Eq. (A.3) with zero initial conditions leads to the following = 7i,+ S:$i,(ai)fETr2,$2,(02) t r=0 i=0 -* </>!;• = K, - • 2(iu;i^iv){2 - sgn(i)) J fori = 0 where (~) denotes a Fourier transform. Similarly, applying the previous procedure to Eq. (A.2), r OO CO -1 M2(^2.1V2. = ^^$2.(02) EW"li®l.(<»0-EW"2,$2,(o2) (A.5) t i=0 r=0 ^ 2 s = {^ 2 , - • 2 C 2 ^^ 2 ,w )(2 - sgn(s)). Solving for VF 2 , from Eq. (A.4) and (A.5), = Trr-Wx-l^^- - (a.6) where _ W2.(a2).^/ii$i,(ai) 1193 Calculating the second term in brackets in Eq. (A.6), fc4'2.(<22)EW^2,«'2.(a2) E where Plugging Eq. (A.7) into Eq. (A.6), M2<j>2s 1 + CKl + Q;2 1=0 \ I' I. -I — ✓ Coupling force Ps Using the definition of transmitted power in Ref. [7], 1112 ( 0 )) is ni2 = Re[-v^o)f;E[P.F;,(o))] L „_ri "I" 0^1 + P i~0 j=:0 J A i<?^2.p hh Ml, where Re[] denotes the real part of the argument, * is a complex conjugate, and Finally, it is assumed that the modal driving forces, f-^., are uncorrelated. Also, the spectral density function of each modal driving force is assumed to be constant for the finite frequency range of interest: C ( \ f ^PlVl ^ V for i^j. Therefore, Eq. (A.9) becomes 2^2fc^O)^.5^pipi ^ ^ ^2r(^2)o)2r to to (A.IO) (A.ll) 1194 PASSIVE AND ACTIVE CONTROL III Research on Control Law of Active Siispension of a Seven Degree of Freedom Vehicle Model Dr&Prof. YuchengLei Lifen Chen Automobile Engineering Dept, Tong Ji University ,Shang Hai,P.R.of China Abstract In the paper , control law of active suspension is presented , which involves 7-DOF vehicle model for improving control accuracy .The control law involve vehicle running velocity , road power spectrum , suspension stiffness and damping .The control law can be applied to multi-DOF control of active suspension of vehicle . Keywords: Active suspension , control law , Game theory , Modeling , 7 — DOF Vehicle Model. 1. Introduction An individual control system for each wheel by applying the optimum regulation method for the two degrees of freedom is showed in [3] . [4] and [5] also introduce two-DOF feedback control method of active suspension .It is difficult for two- DOF control method to coordinate multi-DOF kinematic distances of entire car . Muti-DOF active control can improve coordination control accuracy of entire car, but high speed of CPU is asked for control and calculation while control law of multi-DOF is got by real-time calculation . And, ride performance and handling performance is inconsistent . For resolving the problem ,the paper holds a new calculation method for optimizing the law that can be programmed for real-time control by table-lookup and not by real-time calculation .So the method and law can not only improve coordination control accuracy ,but also develop control speed. 2. Mathematical Model Vehicle is simplified to turn into 7 DOF model . 7-DOF vibration motion equation can be written as follow [Af ]z+ [c]z+ [kY = \C, ](2+ ]e (1) 1195 Where [M] is mass matrix , [C] is suspension damping matrix , [A’] is suspension stiffness matrix , [C,] is tyre • • damping matrix , \K, ] is tyre stiffness matrix , Z is acceleration matrix ,Z is velocity matrix , Z is 7-DOF displacement matrix, Q is road surface input velocity matrix, Q is road surface input displacement matrix. z=p„z....,zy (2) •• . • -j-J Where Zi is vehicle vertical acceleration, Zi is roll •• •• acceleration , Z 3 is pitch acceleration , Z,- (/ = 4,*«*,7) is four tyres vertical acceleration . 3 General Optimization Method of Control Law Objective function of optimization of control law can generally got by calculating weighted sum of 7-DOF mean square root of acceleration , dynamic deflection and dynamic load .it can be written as follows /=! >1 A =1 Where a-^ (z = !,• • (7 = !,• • •A), r * = h* • •A) is weighted ratio . Where a.. (i=l, * * ' ,7) is 7-DOF mean Zi square root of acceleration , o-jy_ 0=1? ‘ ' is 7-DOF mean square root of dynamic deflection , (k=l, * * ^ A) is 7-DOF mean square root of relative dynamic load .cr.., Zi can be calculated by resolving (1) using numerical method. 4 Result of General Optimization Method of Control Law Optimization result of control law of a truck is got using above method as figure 1 and 2 , its main parameters as follows. Wheel distance is 1.4 meter , axle distance is 2.297 meter , mass is 1121.3 kg , front tyre and axle’s mass is 22.8 kg , rear tyre and axle’s mass is 35.0 kg , X axis’ rotational 1196 inertia is 307.4 kg-ni^ , Y axis’ rotational inertia is 1276.5 kg-rri^ . In fig. 1 and 2 , Cl of RMSMIN and C3 of RMSMIN are respectively front and rear suspension damping of getting minimization of above objective function , it is changing while road surface rough coefficient and automobile velocity V is changing . Cl and C3 also rise when velocity V rises . This is called control law of general optimization method of active suspension in the paper . The result in fig. 1 and fig. 2 has been verified by road test. Fig. 1 front suspension Fig. 2 rear suspension optimization damping optimization damping Simulation result can also verify that ride performance’s increasing (suspension stiffness reducing) will make handling performace reduce . So selecting perfect = is very difficult and inconsistent .The paper advances next game method to try to resolve the inconsistent problem . 5 Game Optimization Method of Control Law Because to select weighted ratio of general optimization method is difficult , the paper advances a new method of optimization of control law — Game Balance Optimization Method 4t is discussed as follows . Game theory method of two countermeasure aspect can be expressed as follows: 1197 — 1 j ^ 0 ,z •— 1, 2, tr (4) ^ hj — I , hj ^0 , y = 1, 2,- • *, WZ 2 where .r^, is probability of selecting R^,R 2 > .of countermeasure R ( where R is acceleration mean square root ) , and h,,h^, . X, is probability of selecting ., of countermeasure H ( where H is mean square root of deflection or handling and satiability ) . It is called hybrid game method while these probability is leaded into the method. Countermeasure R selects in order to get maximization of minimization paying expected value of column vector of paying matrix , and countermeasure H selects hj in order to get minimization of maximization paying expected value of row vector of paying matrix . If rank of paying matrix is x ,R should select r., as follows. max\min\ (5) relative And H should select hj as follows : MAXl MINI j=i ( 6 ) relative hj a.j (z = l,.,y = l,.,^ 2 ) in (5) and (6) is element value of paying matrix , basing vehicle theory it can be got as follows : a.j = C, ! + (7) Where C^,C^ in (7) is coefficient of paying matrix ( The paper orders they is 1 as an example , as Q ,C 2 ’ s real value about very much condition is related to some privacy 1198 problem it can’t be introduced.) . (5),(6) called respectively minimization maximization expected value and maximization minimization expected value can be abbreviated as MAxi^IN^^ )} and MIN ^AX(^ )} . if r.,andhj is got as optimization of countermeasure , it can be wntten as follows : MAX^IN^ ^^Optimization Countermeasure Expected Value ^ MIN A probability association {r^Xj) can be content with optimization expected value as follows : Optimization Countermeasure Expected m^ m-i Value=|] J^aij»r;»hj (9) /=! Writer advances reformation simplex algorithm for resolving the game problem as reference [1] . In the paper the water selects only an example to introduce calculation results as follows because the paper has limited space . 6 Results of Game Optimization Method of Control Law Paying matrix as fig. 3 and fig. 4 , optimization result of control law of a truck is got using above game method as figure 5 and 6 , calculated truck’s main parameters as follows. Wheel distance is 1.23 meter , axle distance is 3.6 meter , mass is 13880.0 kg , front tyre and axle’s mass is 280.0 kg , rear tyre and axle’s mass is 280.0 kg , X axis’ rotational inertia is 1935 kg^m^ , Y axis’ rotational inertia is 710 kg-m- . In fig. 3 and 4 3(IJ) is paying matrix value {i = 1,. ,m, J = 1,.,m,). In fig. 5 and fig. 6 , K1 of RMSMIN and K3 of RMSMIN are respectively front and rear suspension stiffness of getting optimization countermeasure expected value of above game method , it is changing while road surface rough coefficient Q and automobile velocity V is changing . K1 and K3 also rise 1199 when velocity V rises . This is called control law of active suspension . The result in fig. 5 and fig. 6 has been verified by road test. Fig. 3 paying matrix Fig. 4 paying matrix 7 Conclusion The paper introduces two method to get optimization control Fig. 5 front suspension Fig. 6 rear suspension optimization stiffness optimization stiffness law of active suspension , and the control law is verified to ability to be applied to real control of active suspension . This will develop control accuracy and speed of active suspension .Off course , it need being researched further . 8 reference 1 Lei Yucheng , Theory and Engineering Realization of Semi — Active Control of Vehicle Vibration ,Dr. paper , Harbin Institute of Technology,China ,1995 6 . 2 Thompson A.G. , A Suspension Proc. Int of Mechanical 1200 Engr. Vol 185 No.36,970 — 990,553 563 . 3 Lei, S., Fasuda , E. and Hayashi, Y. : “An Experimental Study of Optimal Vibration Adjustment Using Adaptive Control Methods ”, Proc . IMechE Int . Conf . Advanced Suspensions , London , England , (1988) , C433/88 , 119- 124. 4 Kamopp D. ,Active Damping in Road Vehicle Systems , VSD, 12(1983), 291-316. 5 Kamopp D . C ., Grosby M. J. & Harword R. ,Vibration Control Using Semi-Active Force , Generator , Trans . ASME, J . Eng . for Ind . Vol. 96 Ser . B , No .2 , (1974), 619-626. 1201 1202 Designing Heavy Truck Suspensions for Reduced Road Damage Mehdi Ahmadian Edward C. Mosch Jr. Department of Mechanical Engineering Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061-0238; USA (540) 231-4920/-9100(fax) ahmadian@vt.edu ABSTRACT The role of semiactive dampers in reducing tire dynamic loading is examined. An alternative to the well-known skyhook control policy, called “groundhook,” is introduced. Using the dynamic model of a single suspension, it is shown that groundhook semiactive dampers can reduce tire dynamic loading, and potentially lessen road damage, for heavy trucks. INTRODUCTION The main intent of this work is to determine, analytically, the role of semiactive suspension systems in reducing tire dynamic loading, and road and bridge damage. Although primary suspension systems with semiactive dampers have been implemented in some vehicles for improving ride and handling, their impact on other aspects of the vehicle remain relatively unknown. Specifically, it is not yet known if implementing semiactive dampers in heavy truck suspension systems can reduce the tire dynamic forces that are transferred by the vehicle to the road. Reducing dynamic forces will result in reducing pavement loading, and possibly road and bridge damage. The idea of semiactive dampers has been in existence for more than two decades. Introduced by Karnopp and Crosby in the early 70’s [1-2], semiactive dampers have most often been studied and used for vehicle primary suspension systems. A semiactive damper draws small amounts of energy to operate a valve to adjust the damping level and reduce the amount of energy that is transmitted from the source of vibration energy (e.g., the axle) to the suspended body (e.g., the vehicle structure). Therefore, the force generated by a semiactive damper is directly proportional to the relative velocity across the damper (just like a passive damper). Another class of dampers that is usually considered for vibration control is fully active dampers. Active dampers draw 1203 relatively substantial amounts of energy to produce forces that are not necessarily in direct relationship to the relative velocity across the damper. The virtues of active and semiactive dampers versus passive dampers have been addressed in many studies [3-10]. Using various analytical and experimental methods, these studies have concluded that in nearly all cases semiactive dampers reduce vibration transmission across the damper and better control the suspended (or sprung) body, in comparison to passive dampers. Further, they have shown that, for vehicle primary suspension systems, semiactive dampers can lower the vibration transmission nearly as much as fully active dampers; without the inherent cost and complications associated with active dampers. This has led to the prototype application, and production, of semiactive dampers for primary suspensions of a wide variety of vehicles, ranging from motorcycles, to passenger cars, to bus and trucks, and to military tanks, in favor of fully active systems. Although there is abundant research on the utility of semiactive dampers for improving vehicle ride and handling, their potential for reducing dynamic forces transmitted to the pavement remains relatively unexplored. This is because most suspension designers and researchers are mainly concerned with the role of suspension systems from the vehicle design perspective. Another perspective, however, is the effect of suspension systems on transmitting dynamic loads to the pavement. ROAD DAMAGE STUDIES Dynamic tire forces, that are heavily influenced by the suspension, are believed to be an important cause of road damage. Cole and Cebon [11] studied the design of a passive suspension that causes minimum road damage by reducing the tire force. They propose that there is a stronger correlation between the forth power of the tire force and road damage than the dynamic load coefficient (DLC) and road damage. A simple measure of road damage, introduced by Cebon m [12], is the aggregate fourth power force defined as Na Al=Y,Pjk k= 1,2,3 ..ns (1) where Pjk = force applied by tire j to point k along the wheel path, ria - number of axles on vehicle, and Us = number of points along the road.. 1204 DLC is a popular measurement frequently used to characterize dynamic loading and is defined as the root mean square (RMS) of the tire force divided by the mean tire force, which is typically the static weight of the vehicle. The equation takes the form; RMS Dynamic Tire Force Static Tire Force This study shows that minimum road damage, for a two degree-of-freedom model, is achieved by a passive system with a stiffness of about one fifth of current air suspensions and a damping of about twice that typically provided. In practice, however, reducing the suspension stiffiiess can severely limit the static load carrying capacity of the suspension and cause difficulties in vehicle operation. Further, higher damping can substantially increase vibration transmission to the body and worsen the ride. In another study by Cole and Cebon [13] a two-dimensional articulated vehicle simulation is validated with measurements from a test vehicle. The effect of modifications to a trailer suspension on dynamic tire forces are investigated. The RMS of dynamic loads generated by the trailer are predicted to decrease by 31 per cent, resulting in a predicted decrease in theoretical road damage of about 13 per cent. Yi and Hedrick compared the effect of continuous semiactive and active suspensions and their effect on road damage using the vehicle simulation software VESYM [14]. A control strategy based on the tire forces in a heavy truck model is used to show that active and semiactive control can potentially reduce pavement loading. They, however, mention that measuring the tire forces poses serious limitation in practice. The primary purpose of this paper is to extend past studies on semiactive suspension systems for reducing road damage. An alternative semiactive control policy, called "groundhook,” is developed such that it can be easily applied in practice, using existing hardware for semiactive suspensions. A simulation model representing a single primary suspension is used to illustrate the system effectiveness. The simulation results show that groundhook control can reduce the dynamic load coefficient and fourth power of tire force substantially, without any substantial increase in body acceleration. 1205 MATHEMATICAL FORMULATION We consider a model representing the dynamics of a single primary suspension in a heavy truck, as shown in Figure 1. Truck Mass Suspension Stiffness Suspension and Tire Mass Tire Stiffness Suspension Damping Road Input Figure 1. Primary Suspension Model This model has been widely used in the past for automobile applications, due to its effectiveness in analyzing various issues relating to suspensions. Although it does not include the interaction between the axles and the truck frame dynamics, the model still can serve as an effective first step in studying fundamental issues relating to truck suspensions. Follow up modeling and testing, using a full vehicle, is needed to make a more accurate assessment. The dynamic equations for the model in Figure 1 are: M^x^+C(x^-X 2 ) +K(x^~X 2 ) = 0 (3a) MjX, -C(ii -X2)-K{x^ -^ 2 ) + 0 The variables Xi and X 2 represent the body and axle vertical displacement, respectively. The variable Xjn indicates road input, that is assumed to be a random input with a low-pass (0 - 25 Hz) filter. The amplitude for Xjn is adjusted such that it creates vehicle and suspension dynamics that resembles field measurements. Such a function has proven to sufficiently represent actual road input to the vehicle tires. Table 1 includes the model parameters, that are selected to represent a typical laden truck used in the U.S. The suspension is assumed to have a linear stiffness in its operating range. The damper characteristics are modeled as a non-linear function, as shown in Figure 2. 1206 Table 1. Model Parameters Description Symbol Value Body Mass Ml 287 kg Axle Mass M2 34 kg Suspension Stiffness K1 196,142 N/m Tire Stifftiess Kt 1,304,694 N/m Suspension Damping C See Table 2 The bilinear function in Figure 2 represents the force-velocity characteristics of an actual truck damper. The parameters selected for both passive and semiactive dampers are shown in Table 2. These parameters are selected based on truck dampers commonly used in the U.S. Although we examined the effect of damper tuning on dynamic loading, if falls outside the scope of this paper. Instead, we concentrate here on comparing different semiactive dampers with a passive damper, using the baseline parameters shown in Table 2. The semiactive and passive damper characteristics used for this study are further shown in Figure 3. 1207 Force Velocity Curve Figure 3. Passive and Semiactive Damper Characteristics SKYHOOK CONTROL POLICY As mentioned earlier, the development of semiactive dampers dates back to early 70’s when Kamopp and Crosby introduced the skyhook control policy. For the system shown in Figure 1, skyhook control implies X,(X] - ;C2) - 0 C = Con X](X^-X2)<0 C = Coff Where x, and represents the velocities of Mi (vehicle body) and M 2 (axle), respectively. The parameters Con and Coff represent the on- and off-state of the damper, respectively, as it is assumed that the damper has two damping levels. In practice, this is achieved by equipping the hydraulic damper with an orifice that can be driven by a solenoid. Closing the orifice increases damping level and achieves Con, whereas opening it gives Coff. 1208 Table 2. Damper Parameters Passive Semiactive On-State Semiactive Off-State n 0.25 0.35 0.03 m 0.10 0.15 0.03 Vbb 0.254 m/sec 0.254 m/sec 0.254 m/sec B 0.20 0.30 0.03 S4 0.10 0.15 0.03 Vbj 0.254 m/sec 0.254 m/sec 0.254 m/sec The switching between the two damper states, shown in Eq. (4), is arranged such that when the damper is opposing the motion of the sprung mass (vehicle body), it is on the on-state. This will dampen the vehicle body motion. When the damper is pushing into the body, it is switched to the off- state to lower the amount of force it adds to the body. Therefore, a semiactive damper combines the performance of a stiff damper at the resonance frequency, and a soft damper at the higher frequencies, as shovm in Figure 4. 1209 Figure 5. Groundhook Damper Configurations: a) optimal groundhook damper configuration, b) semiactive groundhook damper configuration. This feature allows for a better control of the vehicle body, as has been discussed in numerous past studies. The skyhook control policy in Eq. (4), however, works such that it increases axle displacement, X 2 , (commonly called wheel hop). Because the tire dynamic loading can be defined as DL = KtX2 (5) The skyhook control actually increases dynamic loading. As mentioned earlier the development of skyhook policy was for improving ride comfort of the vehicle, without losing vehicle handling. Therefore, the dynamic loading of the tires was not a factor in the control development. GROUNDHOOK CONTROL POLICY To apply semiactive dampers to reducing tire dynamic loading, we propose an alternative control policy that can be implemented in practice using the same hardware needed for the skyhook policy. To control the wheel hop, this policy, called “groundhook,” implies: X, (;ci - X 2 ) < 0 C = Con (6a) x, (Xt - ^2) > 0 C = Coff (6b) As shovm in Figure 5, the above attempts to optimize the damping force on the axle, similar to placing a damper between the axle and a fictitious ground (thus, the name “groundhook”). The groundhook semiactive damper maximizes the damping level (i.e., C = Con) when the damper force is opposing 1210 the motion of the axle; otherwise, it minimizes the damping level (i.e., C = Coff). The damper hardware needed to implement groimdhook semiactive is exactly the same as the skyhook semiactive, except for the control policy programmed into the controller. SIMULATION RESULTS The model shown in Figure 1 is used to evaluate the benefits of groundhook dampers versus passive and skyhook dampers. A non-linear damper model was considered for the simulations, as discussed earlier. The road input was adjusted such that the dynamic parameters for the passive damper resembles actual field measurements. Five different measures were selected for comparing the dampers: • Dynamic Load Coefficient (DLC) • Fourth Power of the tire dynamic load • Sprung mass acceleration • Rattle Space (relative displacement across the suspension) • Axle Displacement, relative to the road Dynamic load coefficient and fourth power of tire force are measures of pavement dynamic loading and are commonly used for assessing road damage. They are both considered here because there is no clear consensus on which one is a better estimate of road damage. Axle displacement, relative to the road, indicates wheel hop and is directly related to DLC and tire force, therefore it is yet another measure of road damage. Sprung mass acceleration is a measure of ride comfort. Our experience, however, has shown that for trucks this may not be a reliable measure of the vibrations the driver feels in the truck. The relative displacement across the dampers relates to the rattle space, that is an important design parameter in suspension systems, particularly for cars. For each of the above measures, the data was evaluated in both time (Figures 6-7) and frequency domain (Figures 8 - 10). In time domain, the root mean square (RMS) and maximum of the data for a five- second simulation are compared. In frequency domain, the transfer function between each of the measures and road displacement is plotted vs. frequency. The frequency plots highlight the effect of each damper on the body and axle resonance frequency. Figures 6 and 7 show bar charts of root mean square (RMS) and maximum time data, respectively. In each case the data is normalized with respect to the performance of passive dampers commonly used in trucks. Therefore, values below line 1.0 can be interpreted as an improvement over the existing dampers. As Figures 6 and 7 show, groundhook dampers significantly 1211 improve pavement loading, particularly as related to the fourth power of tire force. Furthermore, the rattle space is improved slightly over passive dampers, indicating that groundhook dampers do not impose any additional burden on the suspension designers. One measure that has increased due to groundhook dampers is body acceleration. As mentioned earlier, in automobiles this measure is used as an indicator of ride comfort. In our past testing, however, we have found that for trucks it is a far less accurate measure of ride comfort. This is mainly due to the complex dynamics of the truck frame and the truck secondary suspension. A more accurate measure of ride comfort is acceleration at the B-Post (the post Tire Axle Body Rattle Space Dynamic Displacment Acceleration Force ^ 4 Figure 6. RMS Time Data Normalized with respect to Passive Damper Tire Axle Body Rattle Space Dynamic Displacment Acceleration Force 4 Figure 7. Max. Time Data Normalized with respect to Passive Damper 1212 behind the driver), which cannot be evaluated from the single suspension model considered here. Nonetheless, the body acceleration is included for the sake of completeness of data. The model shows that skyhook dampers actually increase the measures associated with pavement loading, while improving body acceleration. This agrees with the purpose of skyhook dampers that are designed solely for improving the compromise between ride comfort and vehicle handling. The improvement in ride comfort occurs at the expense of increased pavement loading. Figure 8. Transfer Function between Axle Displacement and Road Input Figures 8-10 show the frequency response of the system due to each damper. In each figure, the transfer fimction between one of the measures and input displacement is plotted vs. frequency. These plots highlight the impact of skyhook and groundhook on the body and wheel hop resonance, relative to existing passive dampers. The frequency plots indicate that the 1213 Transfer Frequency (Hz) groundhook dampers reduce axle displacement and fourth power of tire dynamic force at wheel hop frequency. At body resonance frequency, groundhook dampers do not offer any benefits over passive dampers. The frequency results for body acceleration and rattle space are similar to those discussed earlier for the time domain results. The frequency plots show that the skyhook dampers offer benefits over passive dampers at frequencies close to the body resonance frequencies. At the higher frequencies, associated with wheel hop, skyhook dampers result in a larger peak than either passive or groundhook dampers. This indicates that skyhook dampers are not suitable for reducing tire dynamic loading. CONCLUSIONS An alternative to skyhook control policy for semiactive dampers was developed. This policy, called “groundhook,” significantly improves both dynamic load coefficient (DLC), and fourth power of tire dynamic load, therefore holding a great promise for reducing road damage to heavy trucks. The dynamic model used for assessing the benefits of groundhook dampers represented a single suspension system. Although the results presented here show groundhook dampers can be effective in reducing tire dynamic loading and pavement damage, more complete models and road testing are necessary for more accurately assessing the benefits. REFERENCES 1. Crosby, M. J., and Karnopp, D. C., "The Active Damper," The Shock and Vibration Bulletin 43, Naval Research Laboratory, Washington, D.C., 1973. 2. Karnopp, D. C., and Crosby, M. J., "System for Controlling the Transmission of Energy Between Spaced Members," U.S. Patent 3,807,678, April 1974. 3. Ahmadian, M. and Marjoram, R. H., “Effects of Passive and Semi-active Suspensions on Body and Wheelhop Control,” Journal of Commercial Vehicles, Vol. 98, 1989, pp. 596-604. 4. Ahmadian, M. and Marjoram, R. H., “On the Development of a Simulation Model for Tractor Semitrailer Systems with Semiactive Suspensions,” Proceedings of the Special Joint Symposium on Advanced Technologies, 1989 ASME Winter Annual Meeting, San Francisco, California, December 1989 (DSC-Vol. 13). 5. Hedrick, J. K., "Some Optimal Control Techniques Applicable to Suspension System Design," American Society of Mechanical Engineers, Publication No. 73-ICT-55, 1973. 1215 6. Hac, A., "Suspension Optimization of a 2-DOF Vehicle Model Using Stochastic Optimal Control Technique," Journal of Sound and Vibration, 1985. 7. Thompson, A. G., "Optimal and Suboptimal Linear Active Suspensions for Road Vehicles," Vehicle System Dynamics, Vol. 13,1984. 8. Kamopp, D., Crosby, M. J., and Harwood, R. A., "Vibration Control Using Semiactive Force Generators," American Society of Mechanical Engineers, Journal of Engineering for Industry, May 1974, pp. 619-626. 9. Krasnicki, E. J., "Comparison of Analytical and Experimental Results for a Semiactive Vibration Isolator," Shock and Vibration Bulletin, Vol. 50, September 1980. 10. Chalasani, R.M., "Ride Performance Potential of Active Suspension Systems-Part 1: Simplified Analysis Based on a Quarter-Car Model," proceedings of 1986 ASME Winter Annual Meeting, Los Angeles, CA, December 1986. 11. Cole, D. J. and Cebon, D., “Truck Suspension Design to Minimize Road Damage,” Proceedings of the Institution of Mechanical Engineers, Vol. 210, D06894, 1996, pp. 95-107. 12. Cebon, D., “Assessment of the Dynamic Forces Generated by Heavy Road Vehicles,” ARRB/FORS Symposium on Heavy Vehicle Suspension Characteristics, Canberra, Australia, 1987. 13. Cole, D. J. and Cebon, D., “Modification of a Heavy Vehicle Suspension to Reduce Road Damage,” Proceedings of the Institution of Mechanical Engineers, Vol. 209, D03594, 1995. 14. Yi, K. and Hedrick, J. K., “Active and Semi-Active Heavy Truck Suspensions to Reduce Pavement Damage,” SAE SP-802, paper 892486, 1989. 1216 Active Vibration Control of Isotropic Plates Using Piezoelectric Actuators A. M. Sadri", J. R., Wright* and A. S. Cherry* The Manchester School of Engineering, Manchester M13 9PL, UK and R. J. Wynnes Sheffield Hallam University, School of Engineering, Sheffield, UK Abstract: Theoretical modelling of the vibration of plate components of a space structure excited by piezoelectric actuators is presented. The equations governing the dynamics of the plate, relating the strains in the piezoelectric elements to the strain induced in the system, are derived for isotropic plates using the Rayleigh-Ritz method. The developed model was used for a simply supported plate. The results show that the model can predict natural frequencies and mode shapes of the plate very accurately. The open loop frequency response of the plate when excited by the patch of piezoelectric material was also obtained. This model was used to predict the closed loop frequency response of the plate for active vibration control studies with suitable location of sensor-actuators. Introduction Vibration suppression of space structures is very important because they are lightly damped due to the material used and the absence of air damping. Thus the modes of the structure must be known very accurately in order to be affected by the controller while avoiding spillover. This problem increases the difficulty of predicting the behaviour of the structure and consequently it might cause unexpected on-orbit behaviour. These difficulties have motivated researchers to use the actuation strain concept. One of the mechanisms included in the actuation strain concept is the piezoelectric effect whereby the strain induced through a piezoelectric actuator is used to control the Research Student, Dynamics &: Control Research Group. ^ Professor, Dynamics & Control Research Group, ^ Former Lecturer, Dynamics & Control Research Group. ^ Professor of Mechanical and Control Engineering. 1217 deformation of the structure [1]. It can be envisaged that using this concept in conjimction with control algorithms can enhance the ability to suppress modes of vibration of flexible structures. Theoretical and experimental results of the control of a flexible ribbed antenna using piezoelectric materials has been investigated in [2]. An active vibration damper for a cantilever beam using a piezoelectric polymer has been designed in [3]. In this study, Lyapunov's second or direct method for distributed-parameter systems was used to design control algorithms and the ability of the algorithms was verified experimentally. These works have clearly shown the ability of piezoelectric actuators for vibration suppression. However, they have been limited to one dimensional systems. Obviously, there is a need to understand the behaviour of piezoelectric actuators in two dimensional systems such as plates. Vibration excitation of a thin plate by patches of piezoelectric material has been investigated in [4]. Their work was basically an extension of the one dimensional theory derived in [1] to show the potential of piezoelectric actuators in two dimensions. In their studies, it was assumed that the piezoelectric actuator doesn't significantly change the inertia, mass or effective stiffness of the plate. This assumption is not guaranteed due to the size, weight and stiffness of the actuator. Based on this assumption, their model can not predict the natural frequencies of the plate accurately after bonding piezoelectric actuators. Therefore, it is essential to have a more general model of a plate and bonded piezoelectric actuators with various boundary conditions. The model should be able to predict frequency responses because this is fundamental to the understanding of the behaviour of the system for control design purposes. It is the objective of the current study to develop such a modelling capability. Previous work [5, 6, 7] has concentrated on the modelling and control of a cantilever beam. The method used involved bonding piezoelectric material to a stiff constraining layer, which was bonded to the beam by a thin viscoelastic layer in order to obtain both active and passive damping. Then a Rayleigh-Ritz model was developed and used to derive a linearized control model so as to study different control strategies. In the work described in this paper, the method has been extended to the more complex plate problem. The paper introduces a modelling approach based on the Rayleigh-Ritz assumed mode shape method to predict the behaviour of a thin plate excited by a patch of piezoelectric material bonded to the surface of the plate. The model includes the added inertia and stiffness of the actuator and has been used to predict the frequency response of the plate. Suggestions for future work are also included. 1218 Theoretical Modelling In developing the Rayleigh-Ritz model of a plate excited by a patch of piezoelectric material bonded to the surface of the plate, a number of assumptions must be made. The patch of piezoelectric material is assumed to be perfectly bonded to the surface of the plate. The magnitude of the strains induced by the piezoelectric element is a linear fimction of the applied voltage that can be expressed by e:=E: ( 1 ) Here is the piezoelectric strain constant, is the piezoelectric layer thickness and V 33 is the applied voltage. The index 31 shows that the induced strain in the ' 1 ' direction is perpendicular to the direction of poling '3' and hence the applied field. The piezoelectric element thickness is assumed to be small compared to the plate thickness. The displacements of the plate middle surface are assumed to be normal to it due to the bending affects. Figure 1 shows the configuration of the bonded piezoelectric material relative to the surface of the plate. w Figure 1. Configuration of the bonded piezoelectric actuator on the surface of the plate. In figure 1, 4 and 4 are the dimensions of the plate, x,, X 2 , y, and y^ are the boundaries of the piezoelectric element and w, v and w are the displacements in the x , y and z direction, respectively. To derive the equations of motion of the plate based on the Rayleigh-Ritz method, both the strain energy U and kinetic energy T of the plate and the piezoelectric element must be determined. The strain 1219 and kinetic energy result from the deformation produced by the applied strain which is induced by exciting the piezoelectric element. The deformations can be expressed by the combination of the midplane displacement and the deformations resulting from the bending of the plate. Strain Energy The strain energy of the plate and piezoelectric material can be calculated by U = +T;«yY,<y)dVp +T^T„)dVp. (2) where 8 is the inplane direct strain, a is the inplane direct stress, t is the inplane shear stress and y is shear strain. dV shows volume differential and indices p and pe refer to the plate and piezoelectric actuator, respectively. The strains 8^ , 8^ and y can be shown to be For the Plate: du dv d (3) _du dv d^w ^ dy dx dxdy For the Piezoelectric actuator: Ev = du a ^~'dx~ ^ a?* Sv a du av « ■ _ 1 ^ dy dx '"33=S,-H'^33 a dxdy ^ ( 4 ) where _ refers to the strains due to the deformation. The stresses , Gy , y can be expressed as 1220 For the Plate: E„ 1 0 ‘e. ' 1 0 y.vy. 0 0 r.y_ 2 . ( 5 ) For the Piezoelectric actuator: ' 0 ^. II 1 0 Sx-H ^33 1 0 r.y_ 0 0 l-v 2 . . Is ( 6 ) where E is Young modulus and v is Poisson's ratio for the assumed isotropic material. Substituting equations 3, 4, 5 and 6 into 2 yields the strain energy of the plate and piezoelectric actuator. ^ S, + sj + 1 (1 - Dp) dVp - '"''33)' + 2Dpp( e, - HV33){ E, - M.V33 )^ + ^ 1 ciVpp. •'Vp, ^ _ u — - tC ( 7 ) Kinetic Energy To obtain the kinetic energy, the velocity components in x, y and z directions are needed. The velocity components can be calculated by differentiating the displacement components which are dw u— u— z— dx dw ( 8 ) dy w=w. Differentiating equations 8 yields 1221 dw u= u- z — ax dw v= V-z— dy w ( 9 ) where u , y and w are the velocity components in the x, y and z directions respectively. Using these velocity components, the kinetic energies of the plate and piezoelectric actuator are obtained as dw P rw^ + (u-z^f+ (v-z^^]dV^ 9x ay' ( 10 ) where p is the mass density. Equation of Motion The static or dynamic response of the plate excited by the piezoelectric actuator can be calculated by substituting the strain and kinetic energy into Lagrange's equation d dT dT ^ dU dt dq. dq,"^ dq. ( 11 ) where q^ represents the ith generalised coordinate and is the ith generalised force. As there are no external forces (the force applied by 5 \e piezoelectric element is included as an applied strain) or gyroscopic terms and there is no added damping, Lagrange's equation reduces to : dt dq, ^ dq, ( 12 ) Now the equation of motion can be obtained by using the expression obtained for the strain and kinetic energy, and the assumed shape functions for flexural and longitudinal motion 1222 u(x,y,t)~ v(x,y,t)={i,(x,y)y w(x, y, t) = {(|)(j:, y)}^ {?(?)}. (13) Here y , ^ and ([) are the assumed displacement shape and h , f and g are generalised coordinate of the plate response m x, y and z directions. Using the shape functions expressed in equations 13, substituting equations 7 and 10 into equation 12, and including Rayleigh damping yields the equation of motion of the plate in the form where M, C and K are mass, damping and stiffness matrices and P is the voltage-to-force transformation vector. Vector q represents the plate response modal amplitudes and V is the applied voltage. State-Space Equations A model of a structure found via finite element or Rayliegh-Ritz methods results in second-order differential equations of the form [M]{9}+[C]{?}+[if]{?}={P}V (15) Choosing state variables x^_ = q and = i,, equation 1 may be reduced to a state-space representation as follows : q = x =X2 / / / (16) q = X2= ~M-^Kq - M~'Cq + M~'PV. Equations 15 can then be rewritten as r 0 /I 0 1 1 1 1 A. + ^2. .^2_ [,] = [/ 0 ] ( 17 ) where [ ], { } are ignored. It should be noted that the vector q must be multiplied by the shape fimctions to produce the actual displacement. 1223 Results The model was used to investigate the response of a simply supported plate. In order to maintain symmetry of the geometric structure a piezoelectric actuator is assumed to be bonded to both the top and bottom surfaces of the plate. So The symmetry of the elements causes no extension of the plate midplane and the plate deforms in pure bending. In this case the shape functions are assumed to be : \{/(x,y) = 0 ^(x,y) = 0 (18) 41 (X, y) = jinf x)sin( ^ y) A- where m and n are the number of half waves in the x and y directions. The properties of the plate are given in Table 1 and its dimensions are 4 = Q38m , 4 , = Q30/w and 4=i.5876m/K. Tables 2 and 3 show the natural frequencies of the bare plate obtained by the thin plate theory and the RR model, respectively. Since the shape functions used in this example express the exact shape of the simply supported plate, the natural frequencies included in Tables 2 and 3 are very close. In order to excite the plate, a piezoelectric actuator with configuration x, = 0.32 Xj = 0.36 m, = 0.04 OT and = 026 m is used above and below. The natural frequencies of the plate after bonding the piezoelectric actuator to the surface are given in Table 4. The results show an increase in natural frequencies, showing that the added stifness is more important than the added inertia. Table 1: Properties of the plate 207 7870 .292 Table 2 : Plate natural frequencies (rad / s), Thin Plate Theory 1 2 3 4 1 437.5 1246.0 2593.5 4480.0 2 941.4 1749.9 3097.4 4983.9 3 1781.2 2589.7 3937.2 5823.7 4 2957.0 3765.5 5113.0 6999.5 1224 Table 3 : Plate natural frequencies (rad / s), RR Model /n 1 J 2 3 4 1 437.5 1245.9 2593.2 4479.1 2 941.3 1749.7 3097.0 4982.8 3 1781.1 2589.4 3936.6 5822.3 4 2956.7 3764.9 5111.9 6997.4 Table 4 : Plate-Piezo natural frequencies (rad / s) /n 1 2 3 4 1 444.0 1257.1 2611.3 4502.8 2 957.2 1775.6 3182.1 5076.7 3 1854.6 2642.5 4072.3 6029.6 4 3076.6 3933.8 5224.6 7277.2 Initially, the piezoelectric actuator was excited by a constant DC voltage. The result of this action is shown in figure 2 which shows the dominant out of plane displacement around the location of the piezoelectric actuator bonded to the surface of the plate. To show the modes of vibration, the piezoelectric actuator was excited by a voltage with frequencies near to the natural frequencies of modes ( 2 , 2 ) and (1,3). The response in figures 3,4 show that the piezoelectric actuator excited both of these modes. Displacement Distribution (X-Y) 0 0 Figure 2. Static Displacement 1225 Displacement Distribution (X-Y) 0 0 Figure 3. Vibration of the plate, mode (2,2) Displacement Distribution (X-Y) 0 0 Figure 4. Vibration of the plate, mode (3,1) The frequency response of tihe plate at the centre was obtained by exciting the piezoelectric actuator at a range of frequencies between 0 and 4000 rad/s. Figure 5 shows the frequency response of the plate at its centre. The frequency response of the plate at y = 0.5L^ along the x-direction is shown in figure 6. It can be seen that the amplitude of vibration of some modes are very high compared to that of the other modes. Special attention must be given for the suppression of vibration of these modes. 1226 Frequency Response of the Plate Figure 5. Frequency response of the plate at the centre Frequency Response of the Plate 4000 Figure 6. Frequency response of the plate along x-direction The results show that it is possible to predict the frequency response of a plate when it is excited by a patch of piezoelectric material. Consequently, a sensor model can be also added to the model and a signal proportional to velocity fed back to the piezoelectric patch. As a result, the closed loop frequency response of the plate can be obtained theoretically which is very important for active vibration control studies. This also permits the investigation of the optimal 1227 location of the actuators and the study of control algorithms for the best possible vibration suppression before using any costly experimental equipment. For this purpose, two patches of piezoelectric (lOcmxScm and 5 cmx4cm), whose specifications are listed in table 5, were bonded to the surface of the plate in different locations and then the plate was excited by a point force marked by "D" in the figure 7. In figure 7 the dash lines are showing the nodal lines of a simply supported plate up to mode (3,3). Table 5: Properties of the actuator (mm) EJxlO''^N/m')pJkg/m') dJxlO'^'mfv) ^ .2 6.25 7700 -180 .3 An actuator is most effective for control of a particular mode if the sign of the strain due to the modal deflection shape is the same over die entire actuator. Consequently, as can be seen from figure 7, the actuators are placed between the nodal lines and at the points of maximum curvature in order to obtain good damping effect on the modes of interest. Then two accelerometers were located at the center of the location of the actuators, marked by "S" in figure 7, in order to have collocated sensor-actuators. The signals obtained by the accelerometers are integrated and fed back to the actuators separately. Therefore rate feedback was used in this configuration. This leads to the feedback control law V = kq (19) where k is an amplification factor or feedback gain. Substituting equation 19 into equation 17 the closed-loop state-space representation of the system can then be obtained as 0 -M-‘K _J I — 1 M-^(C-kP)_ [^]=[/ 0 ] ( 20 ) 1228 Then the effects of the actuators on vibration suppression were investigated. At first, only the actuator near to the center of the plate was used to suppress the vibration. The effect of this is shown in figure 8 . As can be seen, damping in some modes are improved and some modes are untouched. The second actuator was added to the model to see its effect on modes of vibration. Figure 7. Plate with Bonded Piezoelectric Actuators Frequency Response of the Plate Figure 8. Open and Closed loop Frequency Response of the plate solid line; open loop, dash line : closed loop 1229 Frequency Response of the Plate -50 _2qqI-1-1-1-1-1-1——-1-1 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (rad/s) Figure 9. Open and Closed Loop Frequency Response of The Plate solid line: open loop, dash line : closed loop The open and closed loop frequency response of the plate when excited by the point force and controlled by two actuators is shown in figure 9. As can be seen, significant vibration suppression was obtained in both lower and higher modes. Also, it shows that the place of actuators was successfully chosen. This analysis showed that obtaining reasonable but not necessarily optimal placement of actuators in structures is very important in order to obtain a high level of damping in the modes of interest. Obviously, bonding more than one piezoelectric actuator in suitable locations helps to successfully suppress vibration of the plate. Conclusions A model of an active structure is fundamental to the design of control strategies. It can be used to analyse the system and investigate optimal control strategies without using costly experimental equipment. A Rayleigh-Ritz model has been developed to analyse the behaviour of a thin plate excited by a patch of piezoelectric material. The model has been used for a simply supported plate. It has been shown that the model can predict natural frequencies of the plate alone very accurately. The obtained mode shapes also correspond to the actual mode shapes. The frequency response of the plate can be obtained to show the suitability of the model for control design studies. This study allowed the behaviour of the system in open and closed loop form for active vibration control purposes to be investigated. Two piezoelectric actuators were used to investigate their effectiveness on vibration 1230 suppression of the plate. The analysis showed that the location of two actuators was very important to increase the level of damping in both lower and higher frequency modes. The future work will be to extend the model to analyse a plate with more than two patches of piezoelectric material with optimal configuration, obtained by controllability theory, and independent controller for vibration suppression, and experimental verification of the theoretical analysis. References 1. Crawley, E. F. and de Luis, J., Use of Piezoelectric Actuators as Elements of Intelligent Structures, AlAA Journal, Vol. 25, No. 10, 1987, p. 1373. 2. Dosch, J., Leo, D. and Inman, D., Modelling and control for Vibration Suppression of a Flexible Active structure, AIAA Journal of Guidance, Control and Dynamics, Vol. 18, No. 2,1995, p.340. 3. Bailey, T. and Hubbard J. E. Jr., Distributed Piezoelectric Polymer Active Vibration Control of a Cantilever Beam, AIAA Journal of Guidance, Control and Dynamics, Vol. 8, No. 4,1985, p.605. 4. Dimitriadis, E. K., Fuller, C. R., Rogers C. A., Piezoelectric Actuators for Distributed Vibration Excitation of Thin Plates, Journal of Vibration and Acoustics, Vol. 113, No. 1,1991, p. 100. 5. Azvine, B., Tomlinson, G. R. and Wynne, R. J., Use of Active Constrained Layer Damping for Controlling Resonant, Journal of Smart Materials and Structures, No. 4,1995. 6 . Rongong, J. A., Wright, J. R., Wynne, R. J. and Tomlinson, G. R., Modelling of a Hybrid Constrained Layer/Piezoceramic Approach to Active Damping, Journal of Vibration and Acoustics, To appear. 7. Sadri, A. M., Wynne, R. J. and Cherry, A. S., Modelling and Control of Active Damping for Vibration Suppression, UKACC International Conference on Control' 96, 2-5 September 1996. 8 . Bathe, K,, Finite Element Procedures in Engineering Analysis, Prentice- Hall, Inc., 1982. 9. Blevins, R. D., Formulas for Natural Frequency and Mode Shapes, Van Nostrand Remhold, NY 1979. 10. Thomson, W. T., Theory of Vibration with Applications, Prentice-Hall, Inc., 1988. 1231 Active control of sound transmission into a rectangular enclosure using both structural and acoustic actuators S.M. Kim and MJ. Brennan ISVR, University of Southampton, Highfieid, Southampton, S017 IBJ, UK ABSTRACT This paper presents an analytical investigation into the active control of sound transmission in a ‘weakly coupled’ structural-acoustic system. The system under consideration is a rectangular enclosure having one flexible plate through which external noise is transmitted. Three active control systems classified by the type of actuators are discussed. They are; i) a single force actuator, ii) a single acoustic piston source, and iii) simultaneous use of both the force actuator and the acoustic piston source. For all three control systems the acoustic potential energy inside the enclosure is adopted as the cost function to minimise, and perfect knowledge of the acoustic field is assumed. The results obtained demonstrate that a single point force actuator is effective in controlling well separated plate-controlled modes, whereas, a single acoustic piston source is effective in controlling well separated cavity- controlled modes provided the discrete actuators are properly located. Using the hybrid approach with both structural and acoustic actuators, improved control effects on the plate vibration together with a further reduction in transmitted noise and reduced control effort can be achieved. Because the acoustic behaviour is governed by both plate and cavity-controlled modes in a ‘weakly coupled’ structural-acoustic system, the hybrid approach is desirable in this system. 1. INTRODUCTION Analytical studies of vibro-acoustic systems have been conducted by many investigators to achieve physical insight so that effective active control systems can be designed. It is well established that a single point force actuator and a single acoustic piston source can be used to control well separated vibration modes in structures and well separated acoustic modes in cavities, respectively, provided that the actuators are positioned to excite these modes [1,2]. Active control is also applied to structural-acoustic coupled systems for example, the control of sound radiation from a piate[3-6] and the sound transmission into a rectangular enclosure[7-8]. Meirovitch and 1233 Thangjitham[6], who discussed the active control of sound radiation from a plate, concluded that more control actuators resulted in better control effects. Pan et al[Z] used a point force actuator to control sound transmission into an enclosure, and discussed the control mechanism in terms of plate and cavity- controlled modes. This paper is concerned with the active control of sound transmission into a ‘weakly coupled’ structural-acoustic system using both structural and acoustic actuators. After a general formulation of active control theory for structural- acoustic coupled systems, it is applied to a rectangular enclosure having one flexible plate through which external noise is transmitted. Three active control systems classified by the type of actuators are compared using computer simulations. They are; i) a single force actuator, ii) a single acoustic piston source, and iii) simultaneous use of both the force actuator and the acoustic piston source. For all three control systems the acoustic potential energy inside the enclosure is adopted as the cost function to minimise, and perfect knowledge of the acoustic field is assumed. The effects of each system are discussed and compared, and a desirable control system is suggested. 2. THEORY 2.1 Assumptions and co-ordinate systems Consider an arbitrary shaped enclosure surrounded by a flexible structure and a acoustically rigid wall as shown in Figure 1. A plane wave is assumed to be incident on the flexible structure, and wave interference outside the enclosure between the incident and radiated waves by structural vibration is neglected. Three separate sets of co-ordinates systems are used; Co-ordinate x is used for the acoustic field in the cavity, co-ordinate y is used for the vibration of the structure, and co-ordinate r is used for the sound field outside the enclosure. The cavity acoustic field and the flexible structure are governed by the linear Helmholtz equation and the isotropic thin plate theory[9], respectively. The sign of the force distribution function and normal vibration velocity are set to be positive when they direct inward to the cavity so that the structural contribution to acoustic pressure has the same sign as the acoustic source contribution to acoustic pressure. Weak coupling rather than full coupling is assumed between the structural vibration system and the cavity acoustic system. Thus, the acoustic reaction force on the strucural vibration under structural excitation and the structural induced source effect on the cavity acoustic field under acoustic excitation is neglected. This assumption is generally accepted when the enclosure consists of a heavy structure and a big volume cavity. It is also assumed that the coupled response of the system can be described by finites summations of the 1234 uncoupled acoustic and structural modes. The uncoupled modes are the rigid- walled acoustic modes of the cavity and the in vacuo structural modes of the structure. The acoustic pressure and structural vibration velocity normal to the vibrating surface are chosen to represent the responses of the coupled system. 2.2 Structural-acoustic coupled response The acoustic potential energy in the cavity is adopted as the cost function for the global sound control, which is given by [2] where, and Co respectively denote the density and the speed of sound in air, and /?(x,ft)) is the sound pressure inside the enclosure. The vibrational kinetic energy of the flexible structure, which will be used to judge the control effect on structural vibration, is given by[l] where, p., is the density of the plate material, h is the thickness of the plate. If the acoustic pressure and the structural vibration are assumed to be described by a summation of N and M modes, respectively, then the acoustic pressure at position x inside the enclosure and the structural vibration velocity at position y are given by N P(X,(B) = n=I M ni=l where, the N length column vectors ^ and a consist of the array of uncoupled acoustic mode shape functions and the complex amplitude of the acoustic pressure modes a^^(co) respectively. Likewise the M length column vectors O and b consist of the array of uncoupled vibration mode shape functions 0n,(y) and the complex amplitude of the vibration velocity modes respectively. The mode shape functions \f/^{x) and (l)Jy) satisfy the orthogonal property in each uncoupled system, and can be normalised as follows. V = lwlMdV (5) S,=lfyy)dS ( 6 ) 1235 where, V and S/ are the volume of the enclosure and the area of the flexible structure, respectively. Since mode shape functions are normalised as given by (Eq. 5), the acoustic potential energy can be written as £ =_L^a“a (7) ' 4p„c^ Similarly from (Eq. 2) and (Eq. 6), the vibrational kinetic energy can be written as £ =££^b«b ( 8 ) 2 Where superscript H denotes the Hermitian transpose. For the global control of sound transmission, it is required to have knowledge of the complex amplitude of acoustic pressure vector a for various excitations. The complex amplitude of the n-th acoustic mode under structural and acoustic excitation is given by[9,10] ^ 1 . (^) = 4 (^)(^ (x)‘y(x, co)dv +i//'„ (yMy, j (9) where, s(x,Ci)) denotes the acoustic source strength density function in the cavity volume V”, and w(y,6)) denotes the normal velocity of the surrounding flexible structure on surface 5/-. The two integrals inside the brackets represent the nth acoustic modal source strength contributed from s(x,Q)) and u(y,co ), respectively. The acoustic mode resonance term An(fi)) is given by A,.(co) = - -- ( 10 ) o}--Q)-+j2C„<a„(o where a)„and C„ are the natural frequency and damping ratio of the nth acoustic mode, respectively. Substituting (Eq. 4) into (Eq. 9) and introducing the modal source strength q,: = j ¥n X. 0))dV , then we get 2 / M a „(®) = A,(®) (®) + SO',,™ • b,„((0) y V Hi=i where, C„,,„ represents the geometric coupling relationship between the uncoupled structural and acoustic mode shape functions on the surface of the vibrating structure Sf and is given by[l 1] c,,,„ = lvJy)<t>Jy’0>)ds ( 12 ) If we use L independent acoustic control sources, can be written as /=! \l /=! 1236 where =- j W^ control source strength q,^i{co) having an area of Scjj is defined at Xc,i. Thus, the complex amplitude of acoustic modal pressure vector a can be expressed as a = Z.(D,q,+Cb) (14) 2 whereZa= -^A. The matrix A is a (NxN) diagonal matrix in which each (n,n) diagonal term consists of A„, the (NxM) matrix C is the structural-acoustic mode shape coupling matrix, the (NxL) matrix Dq determines coupling between the L acoustic source locations and the N acoustic modes, the L length vector (jc is the complex strength vector of acoustic control sources, and b is the complex vibrational modal amplitude vector. The {NxN) diagonal matrix Za can be defined as the uncoupled acoustic modal impedance matrix which determines the relationship between the acoustic source excitation and the resultant acoustic pressure in modal co-ordinates of the uncoupled acoustic system. Generally the impedance matrix is symmetric but non-diagonal in physical co¬ ordinates, however the uncoupled modal impedance matrix is diagonal because of the orthogonal property of uncoupled modes in modal co-ordinates. Since the flexible structure in Figure 1 is assumed to be governed by the isotropic thin plate theory, the complex vibrational velocity amplitude of the mth mode can be expressed as[10] = + (15) p,hSf ^ where, again p.v is the density of the plate material, h is the thickness of the plate, is the area of flexible structure. Inside the integral /(y,ty), p""\'^.(0 ), and p‘"'(y,ty) denote the force distribution function, and the exterior and interior sides of acoustic pressure distribution on the surface 5/, respectively. Because of the sign convention used, there is a minus sign in front of (y, (O ). The structural mode resonance term Bm((0) can be expressed as -(0^+j2^„^0),„co where co^ and Cm natural frequency and the damping ratio of m-th mode, respectively. Substituting (Eq. 3) into (Eq. 15), then we get b,niO})=——B,„ico)\ g,,,„ico)+ g^,„i(o)-'£cl„-a,(co) (17) p,hSf \ „=i ; 1237 where, L = L • JSj- •'■i/ and Cl^ = C,„.. ■ If we use K independent point force actuators, the m-th mode generalised force due to control forces, gc.m, can be written by = XI <P,n(y)S(y-ycjc)dsf,j,ico) = X^/.».t/<.-.t(®) k=\ ^ k=] ( 18 ) where 0,„(y)5(y-y.,t)rfS, and the k-th control point force l,{co) JSj is located at y^-it. Thus the modal vibrational amplitude vector b can be expressed as b = Y,(g,+Dff,-C’'a) (19) where Ys = —^— B. P..hSjr The matrix B is a (MxM) diagonal matrix in which each (m,m) diagonal term consists of B,„, is the transpose matrix of C, the (NxK) matrix Df determines coupling between the K point force locations and the M structural modes, gp is the generalised modal force vector due to the primary plane wave excitation, the K length vector fc is the complex vector of structural control point forces, and a is the complex acoustic modal amplitude vector. The (MxM) diagonal matrix Ys can be defined as the uncoupled structural modal mobility matrix which determines the relationship between structural excitation and the resultant structural velocity response in modal co-ordinates of the uncoupled structural system. As with the uncoupled acoustic impedance matrix Za, note that Ys is a diagonal matrix. From (Eq. 14) and (Eq. 19), we get a = (l + Z,CY,C’^)''z,(D,q,+CY,g,+CWc) (20) b = (I + Y,C’'Z,C)'' Y,(gp + Dff, - C^Z.D^q,) (21) Since weakly coupling is assumed i.e. = 0 and Y^C^Z^C = 0 , then we get a = Z,(D,q,+CY,g,+CY,D,f.) (22) b = X(g,+D,f,-C%D,q,) (23) Although the formulation developed above covers fully coupled systems, weak coupling is assumed hereafter for the convenience of analysis. 1238 In order to minimise the sound transmission into the cavity, two kinds of actuators are used: a single point force actuator for controlling the structural vibration of the plate and a single rectangular type acoustic piston source for controlling the cavity acoustic pressure. The rectangular piston source is centred at (1.85,0.15,0) with the area of 0.15m by 0.15m, This location was chosen because the sound pressure of each mode in a rectangular cavity is a maximum at the corners, and thus the control source is placed away from the acoustic nodal planes [2]. For a similar reason, the point force actuator is located at (9/20Li, L 2 / 2 ) on the plate, at which there are no nodal lines within the frequency range of interest. Table 2 shows the natural frequencies of each uncoupled systems and their geometric mode shape coupling coefficients which are normalised by their maximum value. Some of natural frequencies which are not excited by the given incident angie((p = 0°) were omitted. The (m/, m 2 ) and («/, 112 , ns) indicate the indices of the m-th plate mode and the n\h cavity mode, and corresponding the uncoupled natural frequencies of the plate and the cavity are listed. A total 15 structural and 10 acoustic modes were used for the analysis under 300 Hz, and no significant difference was noticed in simulations with more modes. 3.2 Active minimisation of the acoustic potential energy This section considers an analytical investigation into the active control of the sound transmission into the rectangular enclosure in Figure 2. Three active control strategies classified by the type of actuators are considered. They are; i) a single force actuator, ii) a single acoustic piston source, and iii) simultaneous use of both the force actuator and the acoustic piston source. Although the formulation developed in this paper is not restricted to a single actuator, each single actuator was used to simplify problems so that the control mechanisms could be understood and effective guidelines for practical implementation could be established. 3.2.1 control using a single force actuator A point force actuator indicated in Figure 2 is used as a structural actuator and the optimal control strength of the point force actuator can be calculated using (Eq. 26). Figure 3(a) shows the acoustic potential energy of the cavity with and without the control force. To show how this control system affects the vibration of the plate, the vibration kinetic energy of the plate obtained from (Eq. 8) is also plotted in Figure 3(b). On each graph, natural frequencies of the plate and the cavity are marked and ‘o’ at the frequencies, respectively. It can be seen that the acoustic response of uncontrolled state has peaks at both 1239 plate and cavity resonances, and the vibration response of uncontrolled state is governed by the plate resonances only because of ‘weak coupling’. Examining Figure 3(a,b) it can be seen that at the 1st, 2nd, 4th, and 5th plate modes corresponding to 52 Hz, 64 Hz, 115 Hz, and 154 Hz, respectively there is a large reduction of the acoustic potential energy. This is because the sound field at these frequencies is governed by the plate vibration modes, and a single structural actuator is able to control the corresponding vibration mode to minimise sound transmission. The structural actuator reduces sound at cavity-controlled modes as well( especially the 2nd and 3rd cavity modes corresponding respectively to 85 Hz and 170 Hz), however it has to increase plate vibration significantly. It shows that minimisation of the acoustic potential energy does not always bring the reduction of structural vibration, and vice versa. Since a cavity-controlled mode is generally well coupled with several structural modes, a single structural actuator is not able to deal with several vibration modes because of 'control spillover\A\. This is the reason why a single acoustic piston source used in the next section was introduced. However, it is clear that a single point force actuator is effective in controlling a well separated plate-controlled mode provided the actuator is not located close to the nodal line. 3.2.2 control using a single piston force source A single acoustic piston source indicated in Figure 2 is used for controlling the acoustic sound field directly. The optimal control source strength of the acoustic piston source can be determined using (Eq. 26). Figure 4 shows the acoustic potential energy of the cavity and the vibrational kinetic energy of the plate with and without the control actuators. Since a plate-controlled mode is generally coupled with several cavity modes, the control effect of the acoustic source is not effective at plate-controlled modes (e.g. 52 Hz, 64 Hz, 115 Hz etc.). Whereas, it is more able to reduce transmitted sound at the cavity-controlled modes ( e.g. 0 Hz, 85 Hz, 170 Hz and 189 Hz) than the structural actuator. As can be noticed from (Eq. 22), the external incident wave and force excitation have the same sound transmission mechanism, which is vibrating the plate and transmitting sound through the geometric mode shape coupling matrix C. Thus, it can be said that the structural actuator is generally effective in controlling sound transmission. At cavity controlled modes, however, several vibration modes are coupled with an acoustic mode. It means that a single acoustic source is more effective than a single structural actuator since a single actuator is generally able to control only one mode. From the results, it is clear that a single acoustic piston source is effective in controlling well separated cavity-controlled modes. It is interesting that there is not much difference in the vibrational kinetic energy 1240 with and without control state. It means that the acoustic actuator is able to reduce sound field globally without increasing plate vibration. 3.2.3 control using both the piston source and the structural actuator In this section, a hybrid approach, simultaneous use of both the point force actuator and the acoustic piston source, is applied. The optimal strength of the force actuator and the piston source can be obtained from Eq. (26). Figure 5 shows the acoustic potential energy of the cavity and the vibrational kinetic energy of the plate with and without the control actuators. Even at the cavity- controlled modes, it can be seen that a large reduction in the acoustic potential energy is achieved without significantly increasing the structural vibration. In general, more control actuators result in better control effects[6]. However, the hybrid approach with both structural and acoustic actuators in the system does not merely mean an increase in the number of actuators. As demonstrated in the last two sections, a single structural actuator is effective in controlling well separated plate-controlled modes and a single acoustic actuator is effective for controlling well separated cavity-controlled modes. Since the acoustic response is governed by both plate-controlled and cavity-controlled modes, the hybrid control approach can be desirable for controlling sound transmission in a ‘weakly-coupled’ structural acoustic system. To investigate the control efforts of each control system, the amplitude of the force actuator and the source strength of the piston source are plotted in Figure 6. There is a large decrease of the force amplitude at the well separated cavity-controlled modes, e.g. 85 Hz and 170 Hz, by using the both actuators. This trend can also be seen in the case of the piston source strength, especially at the 1st and 2nd structural natural frequency (52 Hz, 64 Hz). By using the hybrid approach, simultaneous use of both actuators, better control effects on the vibration of the plate, the transmission noise reduction and the control efforts of the actuators can be achieved. 4. Conclusion The active control of the sound transmission into a ‘weakly coupled’ structural-acoustic system has been considered. The results obtained demonstrates that a single point force actuator is effective in controlling well separated plate-controlled modes, whereas, a single acoustic piston source is effective in controlling well separated cavity-controlled modes. By using the hybrid approach with both structural and acoustic actuators, improved control effects on the plate vibration, further reduction in sound 1241 transmission, and reduced control efforts of the actuators can be achieved. Since the acoustic behaviour is governed by both plate and cavity resonances, the hybrid control approach can be desirable in controlling sound transmission in a ‘weakly coupled’ structural-acoustic system. References 1. C.R. FULLER, S.J. ELLIOTT and P.A. NELSONActive control of vibration, Academic Press Limited, 1996 2. P.A. NELSON and S.J. ELLIOTT Active control of sound, Academic Press Limited, 1992 3. B.-T. WANG, C.R. FULLER and K. DIMITRIADIS Active control of noise transmission through rectangular plates using multiple piezoelectric or point force actuators Journal of the Acoustical Society of America, 1991, 90(5), 2820-2830. 4. M.E. JOHNSON and S. J. ELLIOTT Active control of sound radiation using volume velocity cancellation. Journal of the Acoustical Society of America, 1995, 98(4), 2174-2186. 5. C.R. FULLER, C.H. HANSEN and S.D. SNYDER Active control of sound radiation from a vibrating rectangular panel by sound sources and vibration inputs: an experimental comparison, Journal of Sound and Vibration, 1991,145(2), 195-215. 6. L. MEIROVrrCH and S. THANGJITHAM Active control of sound radiation pressure, Trans, of the ASMS Journal of Vibration and Acoustics, 1990,112, 237-244. 7. S.D. SNYDER and N. TANAKA On feedforward active control of sound and vibration using vibration error signals, Journal of the Acoustical Society of America, 1993, 94(4), 2181-2193. 8. J. PAN C.H. HANSEN and D. A. BIES Active control of noise transmission through a panel into a cavity : I. analytical study. Journal of the Acoustical Society of America, 1990, 87(5), 2098-2108. 9. P.M. MORSE and K.U. INGARD Theoretical Acoustics, McGraw- Hill, 1968 10. E.H. DOWELL, G.F. GORMAN HI, and D.A. SMITH Acoustoelasticity : general theory, acoustic modes and forced response to sinusoidal excitation, including comparisons with experiment. Journal of Sound and Vibration, 1977, 52(4), 519-542. 11. F. FAHY Sound and Structural Vibration, Radiation, Transmission and Response, Academic Press Limited, 1985 1242 incident plane wave Figure 1 A structural acoustic coupled system with the volume V and its flexible boundary surface S/. Figure 2 The rectangular enclosure with one simply supported plate on the surface Sf on which external plane wave is incident with the angles of (cp = 0°) and (0=45°). Table 1 Material properties Material Density Phase speed Young’s Poisson’s Damping (kg/m^) (m/s) modulus (N/m^) ratio (v) ratio (0 Air 1.21 340 - - 0.01 Steel 7870 - 207x10® 0.292 0.01 1243 Table 2 The natural frequencies and geometric mode shape coupling coefficients of each uncoupled system Order: Plate, 1 2 3 4 ■„ , 5. 7 10 ■c ’Type (1,1) (2,1):: 0,1) : (4,1) (5,1) (6,1) (7,1) Cavity^ ' Freq.- 52 Hz 64 Hz 86Hz 115 Hz 154 Hz 200 Hz ;256Hz 1 (0,0,0) 0 Hz 0.71 0 0.24 0 0.14 0 O.IO 2 (1,0.0) 85 Hz 0 0.67 0 0.27 0 0.17 0 3 (2,0.0) 170 Hz -0.33 0 0.60 0 0.24" 0 0.16 4 (0,0.1) 189 Hz -1.00 0 -0.33 0 -0.20" 0 -0.14 5 (1,0,1) 207.HZ 0 -0.94 0 -0.38 0 -0.24 0 6 (2;0,1) 254 Hz 0.47 0 -0.85 0 -0.34 0 -0.22 7 (3,0,0) 255 Hz 0 -0.40 0 0.57 0 0.22 0 1 oo ISO 200 2SO Frequency (a) the acoustic potential energy of the cavity TSO Frequency (b) the vibrational kinetic energy of the plate(dB ref.= 10 ^ J) Figure 3 Effects of minimising the acoustic potential energy using a point force actuator ( solid line : without control, dashed line : with control ), where **’ and ‘o’ are at uncoupled plate and cavity natural frequencies, respectively. 301 Frequency (a) the acoustic potential energy of the cavity Figure 4 Effects of minimising the acoustic potential energy using an acoustic piston source ( solid line ; without control, dashed line : with control), where and ‘o’ are at uncoupled plate and cavity natural frequencies, respectively. Figure 5 Effects of minimising the acoustic potential energy using both a point force actuator and an acoustic piston source - continued 2 Frequency (b) the vibrational kinetic energy of the pIate(dB ref=10‘^ J) Figure 5 Effects of minimising the acoustic potential energy using both a point force actuator and an acoustic piston source ( solid line : without control, dashed line: with control ), where and ‘o’ are at uncoupled plate and cavity natural frequencies, respectively. Frequency (a) the strength of the force actuator Frequency (b) the strength of the piston source( unit: mVsec) Figure 6 Comparison of control efforts of the three control strategies; using each actuator separately ( solid line ) and using both the force actuator and the piston source (dashed line ) , where and ‘o’ are at uncoupled plate and cavity natural frequencies, respectively. 1246 A DISTRIBUTED ACTUATOR FOR THE ACTIVE CONTROL OF SOUND TRANSMISSION THROUGH A PARTITION TJ. Sutton, M.E. Johnson and S.J. Elliott Institute of Sound and Vibration Research University of Southampton, Southampton S017IBJ ABSTRACT The paper considers the problem of active control of soimd transmission through a partition using a single distributed actuator. The use of shaped, distributed actuators rather than point sources or locally-acting piezoceramic elements offers the possibility of controlling the volume velocity of a plate without giving rise to control spillover and avoids an increase in the sound radiated by uncontrolled structural modes. Specifically, a form of distributed piezoelectric actuator is described in which the electrode takes the form of a set of quadratic strips and serves to apply a roughly uniform normal force over its surface. INTRODUCTION The strong piezoelectric properties of the polymer polyvinylidene fluoride (PVDF) were discovered in 1969 [1]. The material is lightweight, flexible, inexpensive and can be integrated into engineering structures for strain sensing and to apply distributed forces and moments for the active control of vibration and sound transmission. Such ^smarf materials offer the possibility of providing lightweight sound-insulating barriers for application to aircraft, ground-based transport and in buildings. Lee [2] has set out the underlying theory of active laminated structures in which one or more layers of flexible piezoelectric material are attached to a plate. Practical sensors using PVDF material have been implemented by Clark and Fuller [3], Johnson and Elliott [4-6], and others. In these cases thin PVDF films were attached to the structure to sense integrated strain over a defined area. In [4] for example a distributed sensor was developed whose output is proportional to the integrated volume velocity over the surface of a plate. 1247 A number of studies have been carried out in which distributed piezoelectric actuators form a layer of a laminated system. In [7] the shape of a distributed piezoelectric actuator was chosen to be orthogonal to all but one of the natural modeshapes of the cylindrical shell system xmder control. Using this approach a set of actuators could be matched to the modes of the system under control, avoiding control spillover {i.e. the excitation of tmcontrolled structural modes). In the present paper a single shaped PVDF actuator is applied to a thin plate to control the noise transmission through it. The shape of the actuator is chosen specifically to apply an approximately uniform force to the plate. Such an actuator can be used to cancel the total volume velocity of the plate and therefore substantially to reduce the radiated sound power. (If volume velocity is measured at the plate surface there is no requirement for a remote error microphone.) As noted by Johnson and Elliott [6], the soimd power radiated by a plate which is small compared with an acoustic wavelength depends mainly on the volume velocity of the plate. The simulations in [6] show that provided the plate is no larger than half an acoustic wavelength, a single actuator used to cancel volume velocity will achieve similar results to a strategy in which radiated power is minimised. It is possible to envisage a large partition made up of a number of active plate elements designed on this basis. CALCULATION OF NORMAL FORCES IN THE PLATE In this section the equation of motion of the plate and attached PVDF layers is set out. The film thickness is assumed to be 0.5 mm. The analysis broadly follows that of Dimitriadis, Fuller and Rogers [8], but the individual piezoelectric coefficients and included separately as is appropriate for PVDF and a sensitivity function is included to account for variations in electrode shape. In addition the bending stiffness of the piezoelectric film is included (it is not negligible as the whole plate is covered). The nomenclature matches that used by Fuller, Elliott and Nelson [9] but here the analysis leads to the inhomogeneous wave equation for the plate-actuator system. We consider an aluminium plate of thickness Ih^ as shown in Figure 1. The plate is covered on its upper and lower surfaces with a piezoelectric film of sensitivity (}>(x,y)d-p in which d.^ is the strain/electric field matrix of the material (3x6 array) and <^{x,y) is a spatially-varying sensitivity function (0 < (j)(x,y) < 1). The two piezoelectric films are assumed to be identical but the same drive voltage is applied with opposite polarity to the lower film. As a result of this antisymmetric arrangement, the plate is subject to pure bending with no straining of the plate midplane. 1248 In line with other similar calculations [2,9] it is assumed that any line perpendicular to the midplane before deformation will remain perpendicular to it when the plate/PVDF assembly is deformed. As a result, the strain at any point in the assembly is proportional to distance z through it. (z=0 is defined to be on the midplane of the plate as indicated in Figure 1.) The direct and shear strains throughout the whole assembly (e^, Ey, E^y) are then given by [10]: 3^^ d^w (1-3) in which w is the displacement of the midplane in the z direction. The corresponding stresses in the plate (only) follow from Hooke's law as in the standard development for thin plates: p __ p_ = i-v: A ^ p 0'' =—— - 2a+vj (4-6) in which is the Young's modulus and Vp is Poisson's ratio for the plate material. Stress in the piezoelectric film follows from the constitutive equations for the material [2]. The direct and shear stresses for the upper piezo film are designated of of and respectively: 's/ K or"’ = C ^ 1 - CO _^36. in which is the voltage applied across the actuator fibn (thickness /rj and 4 and are the strain/field coefficients for the material. For PVDF d^, = 0, but it is included in the analysis for completeness. As in [2], the stiffness matrix C is given by: 1249 0 c ^pe ^ pe^pe 1 — ^ ^ pe ^ pe ^ pe^pe ^pe 1-v' 1-v^ 0 0 0 ^pe 2(l + v^J ( 8 ) The stresses in the lower piezoelectric layer are designated and , and the form of the expression is similar to the upper layer except that the voltage is applied with reversed polarity: 's/ ^31 V = c + (^{x,y) a Summing moments about the x and y axes for a small element dxby of the plate yields the moment per unit length about the y-axis and My about the x-axis; also the corresponding twisting moments per imit length, My^ and M^=jc!’’zdz+ -III, ~K-K ^‘b (10) h,, -III, ‘‘b+lh, My = j a^yZdz + J + J cf'^zdz -III, -hh-K h (11) h,, -III, l>h+lh, = |<t;,z*+ ja^fhdz (12) -III, -K-K h and My^=M,y. The vertical acceleration at each point of the plate d^w/dt^ is obtained by taking moments about the x and y axes for a small element 6x6y and resolving vertical forces as in standard thin plate theory. If the plate is acted on by some external force per unit area p(x,y)f(t) then the vertical motion of the plate is described by: dx^ dxdy dy^ dt^ = -p(x,y)f(t) (13) 1250 in which m is the mass/area of the plate-film assembly. The equation of motion of the plate complete with attached piezoelectric film is obtained by combining the above equations. For convenience the following constants are defined; n _ '' 3(1 (bending stiffness of plate, thickness 2/i,) (14) 3(1-V^) (bending stiffness due to PVDF) (15) + 2 / 1 ,) The equation of motion of the plate assembly including upper and lower piezoelectric layers is then given by: 3^(l)(^,y) 3>(j:,y) The left-hand side of this equation determines the free response of the plate- film assembly and is recognised as the standard form for a thin plate. The first term on the right-hand side is the assumed externally applied normal force per imit area. The second term on the right-hand side gives the effective normal force per unit area applied to the plate due to the two piezoelectric films driven by a voltage . It is clear that this force depends on the spatially-varying sensitivity which has been assumed for the piezoelectric material. An examination of Eqns. (16) and (17) shows that the normal force applied by the piezoelectric film depends on the sum of the plate thickness and the thickness of one of the film layers. If the film is much thinner than the plate (h^« hy) then the applied force becomes independent of the film thickness and depends only on the plate thickness, the applied voltage, the electrode pattern and the material constants. Eqn. (17) also shows that the normal force is applied locally at all points on the plate. No integration is involved, and so in contrast with a volume velocity sensor designed using quadratic strips [5], the force does not depend on the plate boundary conditions in any way. Furthermore, there is no need to use two films oriented at 90 degrees to cancel 1251 the cross-sensitivity It is also worth noting that no assumption has been made about the modeshapes on the plate. We can create a uniform force actuator by choosing: ^ = constant, and (18) Bx (19) This can approximately be achieved by depositing electrodes in the form of narrow strips whose width varies quadratically in the x-direction. (Note that the x-direction is defined as the direction of rolling of the PVDF material, Le. the direction of for maximum sensitivity.) The form of the electrodes is as shown in Figure 2. With this pattern the sensitivity function takes the form: (t)(x,>') = (20) where is the length of the strip. Thus (t)(-^>^) - ^ at x = 0 and x = (no electrode), while (|)(x, 31 ) = 1 halfway along at x = LJ2 (electrode fully covers the film). APPLICATION TO A THIN ALUMINIUM PLATE If a plane wave of sound pressure level 94 dB (say) is normally incident on a hard surface, it will exert a pressure of 2 Pa rms on that surface. If this incident pressure is counterbalanced by a uniform force actuator applied to a plate, then the plate could in principle be brought to rest. Thus for active control of everyday noise levels the uniform force actuator will need to be able to generate a normal force/area of a few pascal over the surface of the plate. (When the incident wave impinges on the plate at an oblique angle, many natural modes of the plate will be excited and it will not be possible to bring it perfectly to rest with a single actuator; however it will remain possible to cancel the plate volume velocity as explained earlier.) By way of example an aluminium plate of thickness 1 mm will be assumed, with a free surface measuring 300 x 400 mm. Attached to each side is a PVDF film of thickness 0.5 mm. One electrode of each panel would be masked to give quadratic strips of length 300 mm as shown in Figure 2. (The width of the strips is unimportant, but should be significantly smaller than the structural wavelength of modes of interest on the plate.) In this case it turns out that = 6.64 for the plate, and 1252 Dp, = 1.27 for the PVDF film. The piezoelectric constants for the film are typically 4 = 23 X m/V and 4 = 3 X lO’"' m/V, giving Cp, = 3.28x10^ Finally the double derivative of the sensitivity function turns out to be = 88.9 (300 mm strip length) dx^ The bending stiffness of the 1 mm aluminium plate is increased by 20% due to the addition of two layers of PVDF film of thickness 0.5 mm each. The force per unit area due to the actuator is obtained from Eqn. (17): force/area = 4 ,(^ 3 ,+Vp, 4)|^'^3 = 6.96x10’^ V 3 Pa Thus 1000 volt rms would yield a tmiform force/area of close to 7 Pa. This is not an impracticable voltage level, but previous experience at ISVR suggests that care would need to be taken to avoid electrical breakdown through the air between electrodes, or over damp surfaces. ACTIVE CONTROL OF SOUND TRANSMISSION In reference [ 6 ] Johnson and Elliott have presented simulations of the active control of harmonic sound transmitted through a plate using a uniform force actuator. Their actuator might be realised along the lines described in this paper. In the simulatioi\s presented in [ 6 ] the uniform force actuator is used with a matched volume velocity sensor having the same electrode shape [4]. The advantage of this configuration is that the actuator can be used to drive the net volume velocity of the plate to zero without exciting high order structural modes in the process (control spillover). Simulations of a 300x380x1 mm aluminium plate showed that reductions in transmitted sound power of around 10 dB were achievable in principle up to 600 Hz using this matched actuator-sensor arrangement. A further advantage of the distributed matched actuator-sensor pair is that the secondary path through the plate (for active control) is minimum phase [ 6 ], giving good stability characteristics if a feedback control loop is implemented to control random incident soimd for which no reference signal is available. 1253 CONCLUSION A design of distributed piezoelectric actuator has been presented which generates a roughly uniform force over the surface of a plate. An example calculation shows that the design is capable of controlling realistic soimd pressure levels. When used in combination with a matched volume velocity sensor, the actuator-sensor pair will have minimum-phase characteristics and will offer the possiblity of feedback control in which neither a reference signal nor a remote error sensor will be required. ACKNOWLEDGEMENT The financial support of the European Community under the Framework IV programme is gratefully acknowledged. (Project reference: BRPR-CT96-0154) REFERENCES [1] G.M. Sessler (1981) JAcoust Soc Am 70(6) Dec 19811596-1608 Piezoelectricity in polyvinylidene fluoride [2] C.K. Lee (1990) JAcoust Soc Am 87(3) Mar 1990 1144-1158 Theory of laminated piezoelectric plates for the design of distributed sensors/actuators. Part I: Governing equations and reciprocal relationships [3] R.L. Clark and C.R. Fuller (1992) JAcoust Soc Am 91(6) June 1992 3321-3329 Modal sensing of efficient acoustic radiators with polyvinylidene fluoride distributed sensors in active structural acoustic control approaches [4] M.E. Johnson, S.J. Elliott and J.A. Rex (1993) ISVK Technical Memorandum 723. Volume Velocity Sensors for Active Control of Acoustic Radiation [5] M.E. Johnson and S.J. Elliott (1995) Proceedings of the Conference on Smart Structures and Materials 27 Feb-3 Mar 1995, San Diego, Calif. SPIE Vol 2443. Experiments on the active control of sound radiation using a volume velocity sensor [6] M.E. Johnson and S.J. Elliott (1995) JAcoust Soc Am 98(4) Oct 1995 2174- 2186. Active control of sound radiation using volume velocity cancellation [7] H.S. Tzou, J.P. Zhong and J.J. Hollkamp (1994) Journal of Sound and Vibration 177(3) 363-378 Spatially distributed orthogonal piezoelectric shell actuators: theory and applications [8] E.K. Dimitriadis, C.R. Fuller and C.A. Rogers (1991) Transactions of the ASME, Journal of Vibration and Acoustics 113 100-107 Piezoelectric actuators for distributed vibration excitation of thin plates [9] C.R. Fuller, S.J. Elliott and P.A. Nelson (1996) Active Control of Vibration. Academic Press, London. [10] G.B. Warburton (1976) The Dynamical Behaviour of Structures, 2nd Edition. Pergamon Press, Oxford. 1254 layers of PVDF film Figure 1: Schematic diagram of thin plate covered on both sides with a layer of PVDF film Figure 2: Electrode pattern of quadratic strips for uniform-force actuator 1255 1256 CONTROL OF SOUND RADIATION FROM A FLUID-LOADED PLATE USING ACTIVE CONSTRAINING LAYER DAMPING J. Ro, A. Al-Ali and A. Baz Mechanical Engineering Department The Catholic University of America Washington D. C. 20064 Abstract Sound radiation from a vibrating flat plate, with one side subjected to fluid-loading, is controlled using patches of Active Constrained Layer Damping (ACLD). The fluid-structure-controller interaction is modeled using the finite element method. The damping characteristics of the ACLD/plate/fluid system are determined and compared with the damping characteristics of plate/fluid system controlled with conventional Active Control (AC) and/or Passive Constrained Layer Damping (PCLD) treatments. Such comparisons are essential in quantifying the individual contribution of the active and passive damping components to the overall damping characteristics, when each operates separately and when both are combined to interact in unison as in the ACLD treatments. I. INTRODUCTION When a structure is in contact with or immersed in a fluid, its vibration energy radiates into the fluid domain. As a result, there is an observable increase in the kinetic energy of the structure due to the fluid loading. Because of this kinetic energy increase, the natural frequencies of structures which are subjected to fluid-loading decrease significantly compared to the natural frequencies of structures in vacuo. Therefore, through understanding of the interaction between the elastic plate structures and the fluid loading has been essential to the effective design of complex structures like ships and submarine hulls. Lindholm et al. [1] used a chordwise hydrodynamic strip theory approach to study the added mass factor for cantilever rectangular plates vibrating in still water. Fu et al [2] studied the dry and wet dynamic characteristics of vertical and horizontal cantilever square plates immersed in fluid using linear hydroelasticity theory. Ettouney et al [3] studied the dynamics of submerged structures using expansion vectors, called wet modes which are finite series of complex eigenvectors of the fluid- structure system. Recently Kwak [4] presented an approximate formula to estimate the natural frequencies in water from the natural frequencies in vacuo. When the structure and the fluid domains become rather complex, solutions of fluid-structure coupled system can be obtained by finite element 1257 methods. Marcus [5], Chowdhury [6], Muthuveerappan et al. [7] and Rao et al [8] have successfully implemented the finite element method to predict the dynamic characteristics of elastic plates in water. Everstine [9] used both finite and boundary element methods to calculate the added mass matrices of fiilly- coupled fluid-structure systems. The above investigations formed the bases necessary to devising passive and active means for controlling the vibration of as well as the sound radiation from fluid-loaded plates. Passive Constrained Layer Damping (PCLD) treatments have been used extensively and have proven to be effective in suppressing structural vibration as reported, for example, by Jones and Salerno [10], Sandman [11] and Dubbelday [12]. Recently, Gu and Fuller [13] used feed-forward control algorithm which relied in its operation on point forces to actively control the sound radiation from a simply-supported rectangular fluid-loaded plate. In the present study, the new class of Active Constrained Layer Damping (ACLD) treatment is utilized as a viable alternative to the conventional PCLD treatment and Active Constrained (AC) with PCLD treatment (AC/PCLD). The ACLD treatment proposed combines the attractive attributes of both active and passive damping in order to provide high energy dissipation-to-weight characteristics as compared to the PCLD treatment. Such surface treatment has been successfully employed to control the vibration of various structural members as reported, for example, by Shen [14] and Baz and Ro [15]. In this paper, the use of the ACLD is extended to the control of sound radiation from fluid-loaded plates. Finite element modeling of the dynamics and sound radiation of fluid- loaded plates is developed and validated experimentally. Particular focus is placed on demonstrating the effectiveness of the ACLD treatment in suppressing the structural vibration and attenuating the sound radiation as compared to conventional PCLD and AC/PCLD. This paper is organized in five sections. In Section 1, a brief introduction is given. In Section 2., the concepts of the PCLD, ACLD and AC/PCLD treatments are presented. In Section 3, the dynamical and fluid finite element models are developed to describe the interaction between the plate, ACLD and the contacting fluid. Experimental validation of the models are given in Section 4. Comparisons between the theoretical and experimental performance are also presented in Section 4 for different active and passive damping treatments. Section 5, summarizes the conclusions of the present study. 2. CONCEPTS OF PCLD, ACLD AND AC/PCLD TREATMENTS Figures (1-a), (1-b) and (1-c) show schematic drawings of the PCLD, ACLD and AC/PCLD treatments respectively. In Figure (1-a), the plate is treated 1258 Figure (1) - Schematic drawing of different surface treatments (a) PCLD, (b) ACLD and (c) AC/PCLD. with a viscoelastic layer which is bonded directly to the plate. The outer surface of the viscoelastic layer is constrained by an inactive piezo-electric layer in order to generate shear strain y, which results in dissipation of the vibrational energy of the plate. Activating the constraining layer electrically, generates a control force Fp by virtue of the piezo-electric effect as shown in Figure (l-b) for the ACLD treatment . Such control action increases the shear strain to yj which in turn enhances the energy dissipation characteristics of the treatment. Also, a restoring moment Mp=d 2 Fp is developed which attempts bring the plate back to its undeformed position. In the case of AC/PCLD treatment, shown in Figure (l-c), two piezo-films are used. One film is active and is bonded directly to the plate to control its vibration by generating active control (AC) force Fp and moment Mp^djFp. The other film is inactive and used to restrain the motion of the 1259 viscoelastic layer in a manner similar to the PCLD treatment of Figure (1-a). In this way, the AC action operates separately from the PCLD action. This is unlike the ACLD configuration where the active and passive control actions operate in unison. Note that in the ACLD configuration, larger shear strains are obtained hence larger energy dissipation is achieved. Furthermore, larger restoring moments are generated in the ACLD treatments as compared to the AC/PCLD treatments as the moment arm d 2 in the former case is larger than the moment arm d 3 of the latter case. This results in effective damping of the structural vibrations and consequently effective attenuation of sound radiation can be obtained. 3. FINITE ELEMENT MODELING 3.1 Overview A finite element model is presented in this section, to describe the behavior of fluid-loaded thin plates which are treated with ACLD, PCLD and AC/PCLD treatments. 3.2 Finite Element Model of Treated Plates (b) (c) Figure (2) - Schematic drawing of plate with ACLD/AC/PCLD patches. Figure (2) shows a schematic drawing of the ACLD and AC/PCLD treatments of the sandwiched plate which is divided into N finite elements. It is assumed that the shear strains in the piezo-electric layers and in the base plate are negligible. The transverse displacement w of all points on any cross section of the sandwiched plate are considered to be the same. The damping layers are assumed to be linearly viscoelastic with their constitutive equations described by the complex shear modulus approach such that G=G ( 1 +T|i). In addition, the bottom piezo-electric layer (AC) and the base plate are considered to be perfectly bonded together and so are the viscoelastic layer and the top piezo-electric layer. The treated plate elements considered are two-dimensional elements bounded by four nodal points. Each node has seven degrees of fireedom to describe the longitudinal displacements u, and v, of the constraining layer, U 3 and V 3 of the base plate, the transverse displacement w and the slopes w and W y of the deflection line. The deflection vector {5} can be written as: {5} = {u„v„U3, V3,W, W ^W y}"" = [{n,} {N 3 } {Nj {n.} {n,} {N,}^ where {5"} is the nodal deflection vector, {Nj}, {Nj}, {N 3 }, {N 4 }, {N 5 }, {N 5 },,, and {N 5 } y are the spatial interpolating vectors corresponding to u„ v„ U 3 , V 3 , w, w^, and Wy respectively. Subscripts ,x and ,y denote spatial derivatives with respect to x and y. Consider the following energy functional ITp for the treated plate/fluid system: np = I(u-TK+w.-w^+wJdv, ( 2 ) where U is the strain energy, T^ is the kinetic energy, is work done by external forces, Wp is work done by the back pressure inside the fluid domain, is work done by the control forces and moments and V is the volume of the plate. These energies are expressed as follows I T,dv =X ^ 1 1„ (*' = 5{®T['^p]{®') ’ |w,dV=j{5'f{F}, = 5' Kj 5' “ V.l^dxdy „[Ui hi ‘J “ ax^ Ui ‘ hi j “ Sy^J where i=l for ACLD control or i=3 for AC control ( 6 ) and I WpdV ={5'}’"[n ]{p'}. where {p®} is the nodal pressure vector of the fluid element. In the above equation []^], [Mp], {F}, [KJ and [Q] are the plate stiffness matrix, mass matrix, external forces vector, piezo-electric forces and moments matrix and plate/fluid coupling matrix as given in the appendix. In equation ( 6 ), d 3 i 32 are the piezo¬ strain constants in directions 1 and 2 due to voltage applied in direction 3. The voltage is generated by feeding back the derivative of the displacement 5 at critical nodes such that j where is the derivative feedback gain matrix and C is the measurement matrix defining the location of sensors. Minimizing the plate energy fimctional using classical variational methods such that |anp/a{6®}j = 0 leads to the following finite element equation: {[K]-»lM.]){5'}-[n]{p'} = {F} ( 8 ) where co is the frequency and [K] = [Kp] + [K^.] is overall stiffriess matrix. 3.3 Finite Element Model of the Fluid The fluid model uses solid rectangular tri-linear elements to calculate the sound pressure distribution inside the fluid domain and the associated structural coupling effects. The fluid domain is divided into fluid elements. Each of 1262 these elements has eight nodes with one degree of jfreedom per node. The pressure vector is expressed by p = [Nf]{p®} and [NJ is pressure shape function and {p®} is nodal pressure vector. Considering the following functional Elf of fluid domain Craggs [16] where [Kf] and [MJ are the fluid stiffiiess and mass matrices as given in the appendix. Minimization of equation (9) such that {OTf/^lp^}} = 0 yields the fluid dynamics as coupled with the structural vibration: ([K,]-o)^[M,]){p'}-<a^[nf{5'} = {0} (10) The boundary conditions involved are of the form ap/an = 0, at a rigid boundary a p / a n = -pp 9^5 / 9, and p = 0. at a vibrating boundary at a free surface where pf is fluid density. 3.4. Solutions of the Coupled Plate/Fluid System Combining equations (8) and (10) gives ■[K]-co^[m,] -[fi] Ip'l.pl (11) [Kr]-CD^[M,]J Lp'J k At low frequencies, the fluid pressure is in phase with the structural acceleration, i.e. the fluid appears to the structure like an added mass. However, as the frequency increases the added mass effect diminishes and the damping effect, i.e. the pressure proportional to velocity, increases. For an incompressible fluid, the speed of sound c approaches oo, thus the mass matrix of the fluid [MJ vanishes, and equation (11) can be simplified to ■[K]-cd=[m,] -[fl]] rs'] pi (12) _ [KfiJkJ k If the fluid-structure coupled system has free boundary surface, then [KJ is non-singular [Everstin, 1991] and the nodal pressure vector {p®} can be eliminated from equation (12) as follows: {p'}=-o.^[K,nnr{5'} (13) 1263 Hence, equation (12) yields ([K]-ffl^(K]+[Mj)){8') = {F) (14) where [MJ is added virtual mass matrix defined by [Muthuveerappan, 1979] [Mj = [a][K,]-'[£2f (15) Equation (14) only involves the unknown nodal deflection vector {5®} of the structure. When {F}=0, equation (14) becomes an eigenvalue problem, the solution of which yields the eigenvalues and eigenvectors. The nodal pressures can then be obtained from equation (13) when the nodal displacements are determined for any particular loading on plate. 4. PERFORMANCE OF PARTIALLY TREATED PLATES WITH FLUID LOADING In this section, comparisons are presented between the numerical predictions and experimental results of the natural frequencies and damping ratios of a fluid-loaded plate treated with ACLD, PCLD and AC/PCLD. The effect of the Active Control, Passive Constrained Layer Damping and Active Constrained Layer Damping on the resonant frequency, damping ratio, attenuation of vibration amplitude and sound radiation are investigated experimentally. The vibration and sound radiation attenuation characteristics of the fluid-structure coupled system are determined when the plate is excited acoustically with broadband frequency excitation while the piezo-electric layers are controlled with various control gains. The experimental results are compared with the theoretical predictions. 4.1 Experimental Set-up Figures (3-a) and (3-b) show a schematic drawing and finite element mesh of the experimental set-up along with the boundary conditions used to describe the fluid-structure system. The finite element mesh includes: 24 plate-elements and 560 fluid-elements. The coupled system has a total of 815 active degree of freedoms. The aluminum base plate is 0.3m long, 0.2m wide and 0.4inm thick mounted with all its edges in a clamped arrangement in a large aluminum base. The aluminum base with mounting frame sits on top of a water tank. One side of the base plate is partially treated with the ACLD/AC/PCLD and the other side is in contact with water. The material properties and thickness of piezo-electric material and the viscoelastic layer listed in Table (1). The size of the combined piezo-electric and viscoelastic patch occupied one-third of the surface area of the base plate and it is placed in the middle of plate as shown in Figure (3-b). A laser 1264 sensor is used to measure the vibration of the treated plate at node 27 as shown in Figure (3-b). The sensor signal is sent to a spectrum analyzer to determine the frequency content and the amplitude of vibration. The signal is also sent via analog power amplifiers to the piezo-electric layers to actively control the sound radiation and structural vibration. The radiated sound pressure level into the tank is monitored by a hydrophone located at 5.0 cm below the plate center. This position is chosen to measure the plate mode (1, 1) which dominates the sound radiation. The hydrophone signal is sent also to the spectrum analyzer to determine its frequency content and the associated sound pressure levels. Figure (3) - The experimental set-up, (a) schematic drawing, (b) finite element meshes. Table (1) - Physical and geometrical properties of the ACLD treatment Layer Thickness(m) Density (Kg/m^) Modulus(MPa) viscoelastic S.OSxlO’" 1104 30** piezoelectric 28xl0-‘ 1780 * Young’s modulus ** Shear modulus 4.2 Experimental Results Experimental validation of the dynamic finite element model of the ACLD/plate system in air has been presented by Baz and Ro [15] in detail. Close agreement was obtained between the theoretical predictions and the experimental 1265 measurements. The d)mamic finite element model is therefore valid to provide accurate predictions. For the uncontrolled treated plate/fluid system, considered in this study, the experimental results indicate that coupling the plate with the fluid loading results in decreasing the first mode of vibration fi*om 59.475Hz to 10.52Hz. The coupled finite element model predicts the first mode of vibration to decrease firom 57.91Hz to 10.24Hz. The results obtained indicate close agreement between the theory and experiments. Figure (4-a) shows a plot of the normalized experimental vibration amplitudes for the fluid-loaded plate with the ACLD treatment using different derivative feedback control gains. According to Figure (4-a), the experimental results obtained by using the ACLD treatment indicate that amplitude attenuations of 11.36%, 48.25% and 75.69% are obtained, for control gains of 2500, 5000, and 13500, respectively. The reported attenuations are normalized with respect to the amplitude of vibration of uncontrolled plate, i.e. the plate with PCLD treatment. Figures (4-b) display the vibration amplitudes of the plate/fluid system with AC/PCLD treatment at different derivative feedback control gains. The corresponding experimental attenuations of the vibration amplitude obtained are 4.6%, 20.29%, 54.04% respectively. Figure (4) - Effect of control gain on normalized amplitude of vibration of the treated plate, (a) ACLD control and (b) AC/PCLD control. Figures (5-a) and (5-b) show the associated normalized experimental sound pressure levels (SPL) using ACLD and AC/PCLD controllers, respectively. The normalized experimental SPL attenuations obtained using the ACLD controller are 26.29%, 50.8% and 76.13% compared to 10.02%, 24.52% and 53.49% with the AC/PCLD controller for the considered control gains. Table (2) 1266 lists the maximum control voltages for the ACLD and AC/PCLD controllers for the different control gains. It is clear that increasing the control gain has resulted in improving the attenuations of the plate vibration and the sound radiation into the fluid domain. It is evident that the ACLD treatment has produced significant vibration and sound pressure level attenuation as compared to the attenuations developed by the AC/PCLD or PCLD treatments. It is also worth emphasizing that the ACLD treatment requires less control energy than the conventional AC/PCLD treatments to control the sound radiation from the plate. Figure (5) - Effect of control gain on normalized sound pressure level radiated from the treated plate, (a) ACLD control and (b) AC/PCLD control. Table (2) - Maximum control voltage for the ACLD/AC/plate system K. 2500 5000 13500 ACLD 0 21.75 V 31.20V 39.60V AC 50.40V 76.38V Figure (6) shows the mode shapes of the first four modes of the treated plate with and without fluid-loading as obtained experimentally using STARMODAL package. Figure (7) shows the corresponding theoretical predictions of the first four mode shapes. Close agreement is found between experimental measurement and theoretical predictions. Figure (8) presents comparisons between the theoretical and experimental natural frequencies and the loss factor of a plate treated with the ACLD and AC/PCLD for different control gains. Close agreement between theory and 1267 experiment is evident. Note also that increasing the control gain has resulted in increasing the damping ratio for both ACLD and AC/PCLD treatments. The comparisons emphasize the effectiveness of the ACLD treatment in acquiring the large damping ratio to attenuate the structural vibration and sound radiation. Figure (6) - Experimental results of first four mode shapes of treated plate (a) without fluid loading and (b) with fluid loading. Figure (7) - Theoretical predictions of first four mode shapes of treated plate (a) without fluid loading and (b) with fluid loading. 5. SUMMARY This paper has presented theoretical and experimental comparisons between the damping characteristics of plates treated with ACLD and 1268 conventional AC with PCLD treatments. The dynamic characteristics of the treated plates when subjected to fluid loading is determined for different derivative control gains. The fundamental issues governing the performance of this class of smart structures have been introduced and modeled using finite element method. The accuracy of the developed finite element model has been validated experimentally. The effectiveness of the ACLD treatment in attenuating structural vibration of the plates as well as the sound radiated from these plates into fluid domain has also been clearly demonstrated. The results obtained indicate that the ACLD treatments have produced significant attenuation of the structural vibration and sound radiation when compared to PCLD and to AC with PCLD. Such favorable characteristics are achieved with control voltages that are much lower than those used with conventional AC systems. The developed theoretical and experimental techniques present invaluable tools for designing and predicting the performance of the plates with different damping treatments and coupled with fluid loading that can be used in many engineering applications. ♦ PCLD ■ ACLD, K:d=l3500 □ AC, K:d= 13500 • ACLD,Kd=2500 O AC, Kd=2500 A ACLD,Kd=5000 A AC, Kd=5000 Theoretical Natural Frequency (Hz) Theoretical Damping Ratio Figure (8) - Comparison between theoretical predictions and experimental results, (a) natural frequency, (b) damping ratio. ACKNOWLEDGMENTS This work is funded by The U.S. Army Research Office (Grant number DAAH-04-93-G-0202). Special thanks are due to Dr. Gary Anderson, the technical monitor, for his invaluable technical inputs. 1269 REFERENCES 1. Lindholm U. S., Kana, D. D., Chu, W. H. and Abramson, H. N., Elastic vibration characteristics of cantilever plates in water. Journal of Ship Research, 1965,9, 11-22. 2. Fu, Y. and Price, W. G., Interactions between a partially or totally immersed vibrating cantilever plate and the surrounding fluid. Journal of Sound and Vibration, 1987,118(3), 495-513. 3. Ettouney, M. M., Daddazio, R. P. and Dimaggio, F. L., Wet modes of submerged structures - part litheory. Trans, of ASMS, Journal of Vibration and Acoustics, 1992,114(4), 433-439. 4. Kwak, M. K., Hydroelastic vibration of rectangular plates. Trans, of ASME Journal of Applied mechanics, 1996, 63(1), 110-115. 5. Marcus, M. S., A finite-element method applied to the vibration of submerged plates. Journal of Ship Research, 1978, 22, 94-99. 6. Chowdury, P. C., Fluid finite elements for added mass calculations. International Ship Building Progress, 1972,19, 302-309. 7. Muthuveerappan G., Ganesan, N, and Veluswami, M. A., A note on vibration of a cantilever plate immersed in water. Journal of Sound and Vibration, 1979, 63(3), 385-391. 8. Rao, S. N. and Ganesan, N., Vibration of plates immersed in hot fluids. Computers and structures, 1985, 21(4), 111-1%!. 9. Everstine G. C., Prediction of low frequency vibrational frequencies of submerged structures. Trans, of ASME, Journal of Vibration and Acoustics, 1991,113(2), 187-191. 10. Jones, I. W. and Salerno, V. L., The vibration of an internally damped sandwich plate radiating into a fluid medium. Trans, of ASME, Journal of Engineering for Industry, 1965, 379-384. 11. Sandman B. E., Motion of a three-layered elastic-viscoelastic plate under fluid loading. J. of Acoustical Society of America, 1975, 57(5), 1097-1107. 12. Dubbelday, P. S., Constrained-layer damping analysis for flexural waves in infinite fluid-loaded plates. Journal of Acoustical Society of America, 1991, (3), 1475-1487. 13. Gu, Y. and Fuller, C. R., Active control of sound radiation from a fluid- loaded rectangular uniform plate. Journal of Acoustical Society of America, 1993, 93(1), 337-345. 14. Shen, I. Y., Bending vibration control of composite plate structures through intelligent constrained layer treatments. Proc. of Smart Structures and Materials Conference on Passive Damping ed. C. Johnson, 1994, Vol. 2193, 115-122, Orlando, FL. 1270 15. Baz, A. and Ro, J., Vibration control of plates with active constrained layer damping. Journal of Smart Materials and Structures, 1996, 5, 272-280. 16. Craggs, A., The transient response of a coupled plate-acoustic system using plate and acoustic finite elements. Journal of Sound and Vibration, 1971, 15(4), 509-528. APPENDIX 1. Stiffness Matrix of the Treated Plate Element The stiffoess matrix [Kp]; of the ith element of the plate/ACLD system is given by Baz and Ro [15]: (A-1) where [K,l and [KJi denote the in-plane, shear and bending stifj&iesses of the ith element. These stiffoess matrices are given by: j = layer 1,2,and3 (A-2) and [K.,1 = I jjB,]"[Dj,][B,]dxdy j = layer 1,2and3 (A-4) with G 2 denoting the shear modulus of the viscoelastic layer and the matrices [BJ, b1 = :^ \({N2}-{N4)/d + {N,} (n,1 +fNj ’ [Bb] = 2{N5},„_ {Naj [B.] = {n. [®^p] = (N.},. + {N3},.+h{Ns},„ {n.},, + + {N2}^ +{n.) „ +h{N4^, 1271 - 1 0 _i 1 - 0 Ei 1 n and fj, 1 EA 1 n 1-vJ Vj 1 u Vj 1 u I_ ' « '“Vi 0 0 i' _o 0 V. . j=l,2and3 (A-5) where h = (hi-h 3)/2 and d = (h 2 +hi/ 2 +D) with D denoting the distance from the mid-piane of the plate to the interface with the viscoelastic layer. Also, Ij represent the area moment of inertia of the jth layer. 2. Mass Matrix of the Treated Plate Element The mass matrix [Mp]; of the ith element of the plate/ACLD system is given by: (A-6) where [Mjp]j and [M^Jj denote the mass matrices due to extension and bending of the ith element. These matrices are given by [Mi,].=p,h,££({N,}^{N,}+{N3}''{N,})dxdy + p,h,[ [({N3)'"{N3} + {NX{N4))dxdy + ip2h2 £ { +{N,}^{N,})dxdy and [ ], = (p,h, + P 3 h 3 + P 3 h 3 ) £ £ [ N 3 f [ N 3 ]dxdy (A-7) where {NJ = {N,}+{N 3 }+h{Ns },3 and {N,} = {NJ+{N 4 }+h{N 5},3 3. Control Forces and Moments Generated by the Piezo-electric Layer 3.1 The in-plane piezo-electric forces The work done by the in-plane piezo-electric forces {Fp}i of the ith element is given by: i{5'}-{Fp}rhi££%d>'dy (A- 8 ) where j=l for ACLD control or j=3 for AC control. Also, Ojp and Sjp are the in¬ plane stresses and strains induced in the piezo-electric layers. Equation (A- 8 ) reduces to: 1272 ='"•1IKFK] for k=l,4 3.2 The piezo-electxic moments The work done by the piezo-electric moments {Mp}i due to the bending of the piezo-electric layer of the ith element is given by: 5' .MM = hj U,e,Ady (A-10) Where Gjb and are the bending stresses and strains induced in the piezo-electric layers. Equation (A-10) reduces to: Mpxk x.' Mpyk d32 Mp,yk _ 0 fork=l,..,4 (A-11) 4. Stiffness and Mass Matrices of the Fluid Element The stiffness matrix [KJj and mass matrix [MJ^ of the ith element of the fluid system are given by: [B,rNdv (A-12) (A-13) [N,]>,]dv (A-13) where [b^] = [[N^],, [N^] ^ [N^] and c is the sound speed. 5. Coupling matrix of the Treated Plate/Fluid System The coupling matrix [Q] of the interface element of the structure/fluid system can be presented by: [ q ]= f [ {N5}''[Nf]dxdy (A-14) Ja Jb 1273 1274 ANALYTICAL METHODS II DYNAMIC RESPONSE OF SINGLE-LINK FLEXIBLE MANIPULATORS E. Manoach^ G. de Paz^ K. Kostadinov^ and F. Montoya^ ^ Bulgarian Academy of Sciences, Institute of Mechanics Acad. G. Bonchev St. Bl. 4; 1113 Sofia, Bulgaria ^ Universidad de Valladolid, E.T.S.I.L Dpto. IMEIM. C/Paseo del Cauce, s/n 47011- Valladolid, Spain 1. INTRODUCTION The flexible-link manipulators have many advantages over the traditional stiff ones. The requirements for light-weight and energy efficient robotic arms could be naturally satisfied by using flexible manipulators. On the other hand the application of the robotic arm in such activities as positioning in electronic microscopes and disc-drivers, hammering a nail into a board or playing tennis, also forces the modeling and control of the dynamic behavior of flexible link manipulators. In most cases the elastic vibrations which arise during the motion must be avoided when positioning the end point of a robotic arm. These are a part of the reasons that cause a great increase of the publications in this topic in recent years. In most papers the flexible robotic arms are modeled as thin linear elastic beams. In [1-3] (and many others) the Bemouli-Euiler beam theory, combined with finite-element technique for discretization with respect to the space variables is used for modeling and control of single-link flexible manipulators. The same beam theory, combined with mode superposition technique is used in [4]. Geometrically nonlinear beam theories are used in [5,6] for the modeling of a single-link and multi-link flexible robotic manipulators, correspondingly. Taking into account the fact that robotic arms are usually not very thin and that the transverse shear could play an important role for dynamically loaded structures [7] the application of the Bernouli-Euiler beam theory could lead to a discrepancy between the robotic arm behavior and that one described by the model. 1275 The aim of this work is to model the dynamic behavior of a single link flexible robotic arm employing the Timoshenko beam theory, which considers the transverse shear and rotary inertia. The arm is subjected to a dynamic loading. As in [3], the viscous friction is included into the model and slip-stick boundary conditions of the rotating hub are introduced. Besides that, the possibility of the rise of a contact interaction between the robotic arm and the stop (limiting support) is included into the model. The beam stress state is checked for plastic yielding during the whole process of deformation and the plastic strains (if they arise) are taken into account in the model. The numerical results are provided in order to clarify the influence of the different parameters of the model on the response of the robotic arm. 2. BASIC EQUATIONS 2.1. Formulation of the problem The robotic arm - flexible beam is attached to a rotor that has friction and inertia. The beam is considered to be clamped to a rotating hub and its motion consists of two components: “rigid-body” component and a component describing the elastic deflection of the beam (see Figure 1). The motion of the flexible beam is accomplished in the horizontal plane and gravity is assumed to be negligible. Figure 1. Model of one-link flexible manipulator. 1276 Tip of the beam (with attached tip-mass) is subjected to an impulse loading. Stick-slip boundary conditions due to Coulomb friction of the hub are introduced when describing the motion of the beam. In other words, if the bending moment, about the hub axis, due to the impact is lower than the static friction torque threshold then the hub is considered clamped and the beam elastic motion is considered only. When the bending moment exceeds friction torque threshold this boundary condition is removed, allowing rotation of the hub and the arm. When the hub speed and kinetic energy of the beam become again beneath the torque threshold, the hub clamps again. The possibility of the rise of a contact interaction between the robotic and the stop is envisaged. In this case, if the hub angle exceeds the limit value, the robotic arm clamps and a part of the beam goes in a contact with the stop, which is modeled as an elastic foundation of a Vinkler type. In view of the fact that the impact loading and contact interactions are included in the investigations, it is expedient to be considered the rise of plastic strains in the beam. 2.2. Deriving the equations of motion. The total kinetic and potential energy of the rotating hub with the attached beam (described by the Timoshenko beam theory) and a lumped mass at it’s tip can be expressed as follow: Ek =^|pA[u(x,t)] dx + |j„[e(t)]^+tMT[u(l,t)fjEl[(()(x,t)] dx EIl^l +kGA’ dx. V dx dx (1 a,b) In these equation u(x,t) is the total displacement u(x,t) = w(x,t)+x0(t) (2) and w is the transverse displacement of the beam, (p is the angular rotation, 9 is the hub angle, E is the Young modulus, G is the shear modulus. A: is a shear correction factor, p is the material density, Jh is the inertia moment of the hub, A=b*h is the beam cross-section area, h is the thickness, b is the width, I=bh^/12,1 is the length of the beam, Mt is the tip mass, t is the time. 1277 Denoting the work of external forces (applied actuating torque T(t) and the beam loading p(x,t))by W=T(t)e(t)+ \p{x,t)w{,x,t)dx 0 the Hamilton's principle can be applied: j5(E,-Ep)dt+ j5Wdt = 0 (3) (4) Substituting eqns (l)-(3) into eqn (4) after integrating and including damping of the beam material, the viscous friction of the hub and the reaction force of the elastic foundation the following equations of motion can be obtained: . r . X. i 2 x de , ,5^w(l,t) , (J + Jh + M-j -1 ^ j ^2 ^ ^ ^ ^2 i JpAx a^w(x,t) dt^ dx = T(t) EI^-^ + kGA dx^ dw dx dw .d^cp -9 “ <^2 - —2 ~ 0<x</, t>0 (5a-c) dt dt kGA + x- a^0 'St" -R(x,t) = p(x,t) The boundary conditions are: w(0,t)=(p(0,t)=0 dcpgt) Q dx kGA dw(l,t) dx d^w(lt) dt^ ( 6 a-c) and the initial conditions are: wfx, 0) =w^(x), w(x, 0 ) = w° (x), p(x,0)=(f^(x), (p(x, 0 ) = 9 ° (x) e(t) = e(t) = o, t<t,,p or 9(0 = when | 9 (t)| and kinetic energy of the beam falls simultaneously under the Coulomb friction thresholds. In eqns (5) the viscous friction coefficient of the hub is denoted by di, d2 and d^ are damping coefficients of the beam material, J is the moment of inertia of the beam about the motor axis and R(x,t) is the reaction force of the stop disposed from Xj to X 2 (xj<X 2 <l) and modeled as an elastic foundation with Vinkier constant r: (7a-d) (7 e,f) (7e)’ 1278 R(x,t) = r[w(x, t) - (x)] for x j < x < x^; |0l >' 0 for 0 <x<x,,X 2 <x</; iBi < The time when the bending moment about the hub axis exceeds the friction torque threshold is denoted by tsUp^ When t>tsiip this condition is removed (allowing rotation of the hub and the beam) until the moment when the beam clamps again. 2.3. Elastic-plastic relationships The beam stress-strain state is usually expressed in terms of generalized stresses and strains which are function of x coordinate only. As a unique yield criterion in terms of moments and the transverse shear force does not exist according to Drucker [8], the beam cross-section is divided into layers and for each of them the stress state has to be checked for yielding. The relation between the stress vector S = and the strain vector s = |-z ^I ’ generally presented as S=[D]8, where in the case of an elastic material [D] = [D^] = is the elastic matrix and f(z) is a function describing the distribution of the shear strains along the thickness. On the basis of the von Mises yield criterion, the yield surface is expressed After yielding during infinitesimal increment of the stresses, the changes of strains are assumed to be divisible into elastic and plastic parts As = As^ + As^ where As 1279 By using eqns (10), (11) and the associated flow rule [9], following Yamada and others [10], the following explicit relationship between the increments of stresses and strains is obtained . AS = [d‘'’]A8 where [-0'^] is the elastic-plastic matrix: HY\d^] LjaLiVi r^r 1 J 1 J L j\asj 1 - C/2 1— 108] In this equation H is a function of the hardening parameter. For ideal plasticity H is equal to zero, while for a wholly elastic material H -> co . 3. Use of mode superposition. 3.1. Rearrangement of the equation of motion. Let the total time interval T on which the dynamic behavior of the structures is investigated, be divided into sequence of time increments y+J. In the numerical calculations the following dimensionless variables are used: x = xl U w = w//, t =tl /c c = / p and then omitting the bars, and after some algebra, the governing equations can be written in the following form: d'e de - T- + C, - d t^ ' d t --cJ w(l,t)- f(p(x,t)dx =C 3 T(t)+ fxp(x,t)dx tv 0 /Vo / 09 0^9 -I c/w I (15 a-c) 0^W 01V 0^iv 09 dt^ ^ dt 10x ^ dx = -p~G{-G^ where a=12P/h^ , p~kG/E, p=pI/(EA), c,= d^l/(cJ}{), Arp/(JHp) C 3 = P/(c^J}^ ), C4~d49/EI, C 3 =djl/EA, =5/6. The nonlinear force due to the reaction of the foundation is denoted by G^ = R(x,t).l/EA and Gf and G 2 are the components of the so-called non-linear force vector Gp( .G^} which is due to the inelastic strains. It has the presentation (see [7]): 1280 3.2 Mode superposition method The l.h.s. of eqns (15 b,c) is a linear form and therefore the mode superposition method can be used for its solution. As the eigen frequencies and the normal modes of vibrations of an elastic beam do not correspond to the real nonlinear system, these modes are called "pseudo-normal" modes. Thus, the generalized displacements vector v = |a”^ (p, w| is expanded as a sum of the product of the vectors of pseudo-normal modes v^ and the time dependent functions q„(t) as ( 17 ) n The analytically obtained eigen functions of the elastic Timoshenko beam are chosen as basis functions (see APPENDIX). When the tip mass is attached to the beam the eigen functions of the system used in the mode superposition method are preliminary orthogonalized by standard orthogonalization procedure as it is mentioned in the APPENDIX. Substituting eqn (17) into (15 b,c), multiplying by v^(x), integrating the product over the beam length, invoking the orthogonality condition (see (A9)) and assuming a proportional damping for the beam material |(c 4 (pf, +c^wl)rdr the following system of ordinary differential equations (ODE) for 0(t) and qn(t) is obtained: ^(f) + c, (9(0 = Cj 2] [«'»(1) - ]?» ^ » + C3(r(0 + P(0) (18a,b) 9„(o+2f„ffl„9„(o+®,k,(o=-‘y„ -'f„&(o+&!’ In this equations 1 11 V f Y„ = J(p„ 6„ = = jxw,Xx)ck, P(t)= \xp{x,t) dx , 1 1 g;(t)= |G'’(x,t)v„(x)dx, g',(t)= jG'(x.t)w„(x)dx, co„ are the eigen 0 0 frequencies of the elastic clamped beam and are the modal damping parameters. The initial conditions defined by eqns (7 a-d) are transformed also in terms of q„(0), and ?„(0) 9 „( 0 ) = 9 „“, 4 „( 0 ) = 4 °, q°„ = J(w°w„ t-a'VVjdx, (j“„ = j(w"w„+a-'(p°(p„)dx, (19a-d) 0 0 The obtained system of nonlinear ordinary differential equations is a stiff one and it is solved numerically by the backward differential formula method, also called the Gear's method [11]. The rise of plastic strains is taking into account by using an iterative procedure based on the "initial stresses" numerical approach [7]. 4. RESULTS AND DISCUSSION Numerical results were performed for the robotic-arm with the same material and geometrical characteristic as these given in [3] in order to make some comparisons. Model parameters are: E=6.5xl0 Pa ,v=0.2, Cp=2.6xl0 Pa, 1=0.7652, b=0.00642 m, h=0.016 m, p=2590 kgW , Mt=0,153 kg, Jh=0.285 kgm^ (Jh is not defined in [3].) The aim of the computations is to show and clarify the influence of the elastic or elastic-plastic deformation on the motion of the robotic-arm, to demonstrate the effect of the hub friction, slip-stick boundary conditions and the contact interaction between the beam and the stop. 1282 Only impact loading on the beam is considered in this work, i.e. dynamic load p(x,t) and applied torque T(t) are equal to zero. The impact loading is expressed as an initial velocity applied to the tip of the beam 0.95 < x < 1. Nine modes are used in expansion (15) but the results obtained with number of modes greater than nine are practically indistinguishable from these shown here. For all calculations the material damping is equal to 8% of the critical damping. The results for the rotation of the hub of the flexible manipulator with an attached mass at its tip -1 and without an attached mass -2 are shown in Figure 2. The hub friction is not considered. The beam is subjected to an impact loading with initial velocity =-1.95 m/s. As can be expected, the hub angle increases much faster in the case of the beam with an attached mass. The corresponding beam deflections are presented in Figure 3. The results obtained are very close to these obtained in [3] (Fig. 7 and 8 in [3]). The frequencies of forced vibrations obtained in [3], however, correspond to the beam without an attached mass. 0, rad -4.00 I-^^^^-1- ^^i^ 0,00 l.OO 2.00 3.00 4.00 5.00 t, sec Figure 2. Hub response without viscous friction. 1 - beam with an attached mass; 2- without an attached mass w, m Figure 3. Deflection of the tip of the beam without viscous friction of the hub. 1 - beam with an attached mass; 2 - without an attached mass 1283 The influence of the hub friction on the flexible manipulator response can be seen in Figure 4. The viscous friction is set di=0.1 Nms, the static Coulomb friction threshold is equal to 0.06 Nm and three cases of the hub slip-stick threshold are tested: |e| stick = 0-005 rad.s'^ - 1, |0| stick = 0.0085 rad.s'^ , stick = 0.01 rad.s'^ For this initial velocity (w° =-1.95m/s) the hub slips very fast from the initial clamped state and the beam begins to rotate. As can be expected, the consideration of the viscous friction of the hub leads to a decrease of the angle of rotation of the beam and changes the linear variation of 0 with time. The results show also that the value of the hub speed threshold |0| stick exercises an essential influence on the motion of the rotating system. When |e| =0.005 rad.s' the beam sticks at t = 2.602 s after that the hub periodically slips and sticks which also leads to damping of the motion. When joj stick “ 0.0085 rad.s'^ the start of sticking occurs at t = 1.7207 s and after t=2.417 s the hub clamps with short interruptions till t=3.4 s when due to the elastic vibrations it snaps in the direction opposite to w°» clamps again at 3.679s, slip at 4.5s, and finally clamps at t=4.5s. When joj stick =0.0085 rad.s"' the sticking begins at t=1.0525s and very fast (at t«2 s) the beam clamps with 0=-O.587 rad. In order to observe the occurrence of the plastic deformation the beam was subjected to impulse loading having larger values of initial velocities. In addition, the contact interaction between the beam and the stop was considered. The beam-tip deflection in the presence of a contact with the stop disposed at X €[0.16, 0.263] and initial velocity w° =-15.95 m/s is shown in Fig. 5. 0,rad di=0.1Nms. 1- |0| stick”0-005rad.s"', 2 - |0| stick = 0-0085 rad s"' ; 3- |0| stick =0-0is-' 1284 In order to reduce the computational time the limit value of 0 was chosen e^j^p=0.0025 rad. When this value was reached the problem was automatically resolved with new initial conditions 0=0, w°=w(Xstop,tstop), etc. As can be seen, the presence of the contact interaction during the process of motion of the beam due to the elastic support for x e[0.16, 0.263] leads to a decrease of the amplitudes of vibrations in the direction of the stop in comparison with the amplitudes in the opposite direction. The variation of the beam displacements along the beam length for the first 0.8 s of motion is shown in Fig. 6. It must be noted that in this case of loading a plastic yielding occurs. It is assumed that beam material is characterized by an isotropic linear strain hardening and H=0.5. The plastic yielding occurs simultaneously with the contact interaction at t=0.01366 s at the clamped end of the beam. At t=0.0186 s the plastic zone spreads to x = 0.158 and at t=0.08767 if covers the length to X =0.31. The last points that yields are x =0.55, 0.61 at t=l. 119 s. Seven layers along the beam thickness, symmetrically disposed about the beam axis was checked for yielding (N2=7) but the plastic zone has reached the second and 6th layers only at the clamped end of the beam (x =0). In all other point along beam length the plastic yielding occurs only at the upper and lower surface of the beam. The plastic strains are small and the response of the beam is not very different from the wholly elastic response. Nevertheless, the appearance of such kind of plastic deformations in the structures used for the precise operations must be taken into account in the manipulator self calibration procedure. w,m 0.40 0.30 0.20 0,10 0.00 -O.IO - 0.20 -0.30 0.00 1.00 2.00 3.00 4.00 5.00 t, sec. Figure 5. Deflection of the tip of the beam with time in the case of a contact with the stop. r=5.5xl0^ Pa 5. CONCLUSIONS In this work a model describing dynamic behavior of a deformable beam attached to a rotating hub that has friction and inertia is developed. The Timoshenko beam theory is used to model the elastic deformation of the beam. 1285 Figure 6, Variation of the elastic-plastic beam displacement along the beam length with time in the case of a contact with the stop. The slip-stick boundary conditions are also incorporated into the model The possibility of the rise of undesired plastic deformations in the case of a high velocity impact on the clamped robotic arm, or in the case of a contact with limiting support (stop) is included into the model. The analytically obtained eigen functions of the elastic Timoshenko beam vibrations are used to transform the partial differential equations into a set of ODE by using the mode superposition method. This approach minimizes the number of ODE which have to be solved in comparison with another numerical discretization techniques (finite elements or finite difference methods). The results obtained show the essential influence of elasticity on the robotic- arm motion. The model will be used to synthesise a control of one link flexible manipulators and for a self calibration procedure when plastic deformation would occur. Acknowledgments The authors gratefully acknowledge the financial support from EC Copernicus Program under the Project ROQUAL CIPA CT 94 0109. The first author wishes to thank the National Research Fund for the partial financial support on this study through Contract MM-517/95. 1286 REFERENCES 1. Bayo, E. A finite-element approach to control the end-point motion of a single -link flexible Robert. J. Robotic System, 1987, 4„ 63-75 2. Bayo, E. and Moulin, H., An efficient computation of the inverse dynamics of flexible manipulators in the time domain. IEEE Proc Int. Conf. on Robotics and Automations, 1989, 710-15. 3. Chapnik, B.V., Heppler, G.R., and Aplevich, J.D. Modeling impact on a one-link flexible robotic arm. IEEE Transaction on Robotics and Automation, 1991,7,479-88. 4. Liu, L. and Hac, A., Optimal control of a single link flexible manipulator. Advances in Robotics, Mechatronics, and Haptic Interfaces, 1993, DCS- 49, 303-13. 5. Wen, J.T., Repko, M. and Buche, R., Modeling and control of a rotating flexible beam on a translatable base. Dynamics of Flexible Multibody Systems: Theory and Experiment, 1992, DCS-37, 39-45. 6. Sharan, A.M. and Karla, P., Dynamic Response of robotic manipulators using modal analysis. Meek Mach. Theory, 1994, 29, 1233-49. 7. Manoach, E. and Karagiozova, D. Dynamic response of thick elastic-plastic beams. International Journal of Mechanical Sciences, 1993, 35, 909-19 8. Drucker, D.C. Effect of shear on plastic bending of beams. J. of Applied Mechanics, 1956,23, 515-21 9. Hill, R. Mathematical Theory of Plasticity, 1950, Oxford University Press, London. 10. Yamada, Y., Yoshimura, N. and Sakurai T., Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by the finite elements, Int. J. of Mechanical Sciences , 1968,10, 343-54 11. Gear, C.W., Numerical initial value problem in ordinary differential equations, 1971, Prentice-Hall, Englewood Cliffs, NJ. 12. Abramovich, H. Elishakoff, 1. Influence of shear deformation and rotary inertia on vibration frequencies via Love’s equations. J. Sound Vibr., 1990, 137, 516-22. 1287 APPENDIX: NORMAL MODES OF FREE VIBRATIONS OF A CLAMPED TIMOSHENKO BEAM WITH AN ATTACHED MASS, Equations (5 b,c) can be decoupled, transforming them into two fourth order equations [12] as regards cp and w. Solving this equations (with p=0 and R=0) and using the boundary conditions (6) (with 0=0), the equations of the frequencies and forms of vibrations of the beam are obtained. Introducing following denotations 1/2 (A1 a-e) ^2„={®,^(l + P) + [®:a + P)^+4(B,^(a-pco,^,)]''^} /2, /l„ =(4 +CoJP)/5,„, /2„ /3„ =(4-®SP)''-S!». the frequencies of free vibrations are determined as roots of equation: a) In the case >0 i.e. co < a / p the frequencies equation is: ^11^22 ”^12^21 “ ^ > (-^2) where ^11 ~/ 2 n‘^ 2 n ^12 ~ ■^/ 2 «‘^ 2 « ®^^(‘^ 2 «) h\ = -f\nM.Sxn ) “ /2„ sin(52„ ) + s^„sh(si, ) + 52„ sin(52„ ) + , . . (A3 a-d) M‘^lnCh(^lJ + ‘^2« cos(,y2J] *22 = /2j=OS(^2«)-cll('SlJ] + 4^S,„ch(i,„)-52„ COS(52„) + J\n + sl„ sin(i 2 „)] Jin and the modes of vibrations are: w„W = -S„ 1 ' b ^(ch(i,„x) - cos(j2„x)) + sin(i2„x) - ^sh(i|„x) J\» L^i (A4 a,b) (/,„sh(,s,„x) + sin(,y 2 «^)) + fi,, icos{s^„x) - ch(s,„x) b) In the case 5 ^,, <0 i.e. > a / P the eigen frequency equation (A2) has the following presentation: 1288 b\\ = + /2„i2„ C0S(J2„) bn = ?i„/2nSin(?i„) - /2„i2n sin(j2„) 621 = /3„sin(ii„) - /2„ sin(52„) -?i„sin(Ji„) + S2„ sinC^jJ - >.[J,^„COS( j,„ ) - sl„ C 0 S(i 2 „)] *22 = /2„[cOS(S2„) - COS(J,„)] - ^ S,„COS(?,„) + S 2 „ COS{s-^„) - J\n - sl„ sin(52„)] 7 In sh = -5?„ and the of vibrations are: w„(x) = f b sin(j2„2c) - ^sin(?,„x) - ^(cosCij,,*) - cos(j,„x)) /i„ *11 'P„W = -S» (A 5 a-e) (A6a-b) /j,, (cos(i2„x) - cos(^|„x)) - ^(/3„sin(i,„x) - f^„ sin(jj„x)) 0,, When a mass is not attached at the beam tip the following orthogonality condition is fulfilled: 1 p fO, n^ m; (a _ J [1, n - m. (A 7 ) and when an attached mass is considered the modes are orthogonalized by standard orthogonalization procedure. The constants are obtained from condition (A 7 ). 1289 1290 Wave Reflection and Transmission in an Axially Strained, Rotating Timoshenko Shaft B. Kang ‘ and C. A. Tan ^ Department of Mechanical Engineering Wayne State University Detroit, Michigan 48202, U. S. A. Abstract In this paper, the wave reflection and transmission characteristics of an axially strained, rotating Timoshenko shaft under general support and boundary conditions, and with geometric discontinuities are examined. The static axial deformation due to an axial force is also included in the model. The reflection and transmission matrices for incident waves upon these point supports and discontinuities are derived. These matrices are combined, with the aid of the transfer matrix method, to provide a concise and systematic approach for the free vibration analysis of multi-span rotating shafts with general boundary conditions. Results on the wave reflection and transmission coefficients are presented for both the Timoshenko and the simple Euler-Bemoulli models to investigate the effects of the axial strain, shaft rotation speed, shear and rotary inertia. ‘ Graduate Research Assistant. Tel: +1-313-577-6823, Fax: +1-313-577-8789. E-mail: kang@feedback.eng.wayne.edu ’ Associate professor (Corresponding Author),Tel: +1-313-577-3888, Fax:+1-313.577-8789. E-mail: tan@tan.eng.wayne.edu Submitted to: Sixth International Conference on Recent Advances in Structural Dynamics, Institute of Sound and Vibration, Southampton, England, July, 1997 1291 A. NOMENCLATURE Area of shaft cross section [m^] do Diameter of shaft cross section [m] C Generalized coordinate of an incident wave [m] Cdt {Ct) Translational damping coefficient [N-sec/m] Cdr (Cr) Rotational damping coefficient [N-m-sec/rad] Co Bar velocity [m/sec] Cj Shear velocity [m/sec] D Generalized coordinate of a transmitted wave [m] E,G Young’s and shear modulus [NW], respectively I Lateral moment of inertia of shaft [m'^] Jm (Jm) mass moment of inertia of a rotor mass [kg-m"^] K Timoshenko shear coefficient Knikr) Rotational spring [N/rad] Kt (kd Translational spring [N/m] t Length of shaft [m] M{m) Mass of rotor [kg] P Axial force [N] rij, ti] Reflection and transmission coefficients, respectively, i - 1 positive traveling U{u) wave; i = 2 negative traveling wave; jr = 1 propagating wave for Cases 11 and /V; 7 = 2 attenuating wave for Cases II and IV. Both 7 =1, 2 for propagating wave for Case I Transverse displacement [m] X-Y-Zix-y-z) Reference frame coordinates [m] a {K-G)IE Rotation parameter, see Eqn. (Id) £ Pl{E-A), axial strain e' Non-dimensional axial load parameter, see Eqn. (13b) T, f (r, y) Wavenumber [m‘‘] nAi) See Eqns. (20a, b), (22a, b) and (24a, b) p Mass density of shaft [kgW] G Diameter ratio between two shaft elements 57, (ft)) System natural frequency for Timoshenko model [rad/sec] a System natural frequency for Euler-Bernoulli model [rad/sec] Q. Rotation speed of shaft [rad/sec] W (¥) Bending angle of the shaft cross-section [rad] subscript L, r The left and right side of a discontinuity, respectively. superscript -,+ Negative and positive traveling waves, respectively, when used in C and D. Note: Symbols in Otherwise denotes quantities on the left and right side of a discontinuity, respectively parenthesis are the corresponding non-dimensional parameters. 1292 1. INTRODUCTION The vibrations of elastic structures such as strings, beams, and plates can be described in terms of waves propagating and attenuating in waveguides. Although the subject of wave motions has been considered much more extensively in the field of acoustics in fluids and solids than mechanical vibrations of elastic structures, wave analysis techniques have been employed to reveal important, physical characteristics associated with vibrations of structures. One advantage of the wave technique is its compact and systematic approach to analyze complex structures such as trusses, aircraft panels with periodic supports, and beams on multiple supports [1]. Previous works based on wave propagation techniques have been well documented in several books [2-4], Recently, Mead [5] applied the phase-closure principle to determine the natural frequencies of Euler-Bemoulli beam models. A systematic approach including both the propagating and near- field waves was employed to study the free vibrations of Euler-Bemoulli beams [6]. High speed rotating shafts are commonly employed in precision manufacturing and power transmission. Despite the usefulness of the wave propagation method in structural vibrations, applications of this technique to study the dynamics and vibrations of a flexible shaft rotating about its longitudinal axis have seldom been considered. The purpose of this paper is to examine the wave reflection and transmission [6] in an axially strained, rotating Timoshenko shaft under various support and boundary conditions. The effect of the axial load is included by considering the axial static deformations in the equations of motion. This paper is a sequel to another paper in which the authors discuss the basic wave motions in the infinitely long shaft model [7]. Although there have been numerous studies on the dynamics and vibration of rotating shafts, none has examined the effects of axial strains (which cannot be neglected in many applications) on the vibration characteristics of a Timoshenko shaft under multiple supports. Modal analysis technique has been applied to study the vibration of a rotating Timoshenko shaft with general boundary conditions [8, 9], and subject to a moving load [10]. Recently, the distributed transfer function method was applied to a rotating shaft system with multiple, geometric discontinuities [11], The wave propagation in a rotating Timoshenko shaft was considered in Ref. [12]. Other major works on the dynamics of rotating shafts have been well documented in Refs. [13-15]. This manuscript is organized as follows. Governing equations of motion [16] and basic wave solutions for the Timoshenko shaft are outlined in Section 2. Each wave solution consists of four wave components: positive and negative, propagating and attenuating waves. In Section 3, the wave reflection and transmission matrices are derived for the shaft under various point supports and boundary conditions. The supports may include translational and rotational springs and dampers, and rotor mass. Results are presented for both the Timoshenko and the simple Euler- Bemoulli models to assess the effects of axial strain, shaft rotation, shear and rotary inertia. The wave propagation across a shaft with geometric discontinuities such as a change in the cross- section is examined in Section 4, and the wave reflection at a boundary with arbitrary support conditions is considered in Section 5. With the wave reflection and transmission matrices as the main analytical tools, it is shown in Section 6 how to apply the current results together with the transfer matrix method to analyze the 1293 free vibration of a rotating, multi-span Timoshenko shaft system in a systematic manner. The proposed approach is then demonstrated by considering the free vibration of a two-span beam with an intermediate support. 2. FOKMULATION AND WAVE SOLUTIONS Consider a rotating shaft subjected to axial loads and with multiple intermediate supports and arbitrary boundary conditions, as shown in Fig. 1. Including the effects of rotary inertia, shear deformations, and axial deformations due to the axial loads, the uncoupled equations of motion governing the transverse displacement u and the slope \}/ due to bending can be derived in the following non-dimensional form .. . d'^u „ d^u £ +16a(l + £)(l-i-e-)^r^ = 0 , a dr U lA £ . d~U ■2ri3^ + a—-16£{1 + £--)^^ a dz'^ -l-16a(l-l-£)(l + £-—= 0 , a dr u = — z = — r = — T = ^ ■ a.' a.' T. ^ \ KG Figure 1. A rotating Timoshenko shaft model subject to axial loads and with general boundary conditions. 1294 (Id) a KG E ’ E Note that u and y/are the measurements in the complex plane, that is u=ux+iuy and \i/=\}fx+iYr E denotes the Young’s modulus, p the mass density, As the area of the cross section, ao the diameter of shaft, K the Timoshenko shear coefficient, G the shear modulus and Q the constant angular velocity of the shaft. Details of deriving these equations of motion are found in Ref. [16]. Assuming and substituting the following wave solutions into Eqns. (la) and (lb) = ( 2 a) = ( 2 b) and defining the non-dimensionalized wavenumber 7 and system natural frequency w gives the frequency equation, Eqn. (3a); see Ref. [7], y = r^o . _ CO a, KG . (j) = -a. = -is known as the shear velocity). where, 7 “* -A 7 " 4-5 = 0 , A = (1 -i- a)a) ^ - 2j3 cu “ 16£ (1 + e-), 5 = a ot)^ - 2j3 u) - 16a (14-£)(1 4 -e - •^) The four roots of Eqn. (3a) are 7 = ±-^[a ± -x/a^ -45 . ( 2 c) ( 2 d) (3a) (3b) (3c) (4) In general, 7 is complex. Let (o be real. It can be shown that, with a > 0 and £ the axial strain of the elastic solid, the discriminant A^ -45 is positive semi-definite for most engineering applications. Hence, it is possible to classify the wave solutions into four distinct cases. Note that one may study the wave propagation by considering only a single general form of the wave solution. However, the classification procedure identifies the coupled modes of vibration of the 1295 Timoshenko shaft model and provides a better understanding on how each wave solution governs the wave motions [7]. Based on the algebraic relationships between A and B, the four valid wave solutions are obtained as follows. Case / (A > 0 and B > 0); «(z, t) = + C-(5a) V(z, t) = (CJ.e-'*’" + )e® (5b) Case !I{A>0 andB<0): «(z, t) = (Ce-"''' + + C:^e-^'-‘ + (6a) yz(z,t) = (c;,^-'^" - c;,/" + + 0 ;^/=- )£® (6b) CaseIIIiA<0mdB>0): uiz,t) = (C>-''' + + C:,«")«'“ (7a) Vr(z,0 = (C;,a-" + C;,ef>' + + C;je'=')«® (7b) Coje/V(A<0 andB<0): k(zA) = (C,>-^'^ +C,>’‘'- + C>-'''=‘ + C,V'’'*')«'® (8a) V/(z,«) = (C;,e-f- + (8b) where, f, =^(|A| + V-5’-4|B|f, r. (9^. W r, = + 4|B| + |A|)^, r, + 4|B| - |A|j', (9c,d) and the coefficients C* and C" denote positive- and negative-travelling waves from the origin of disturbance, respectively. Important remarks on the basic wave propagation characteristics are summarized from [7]. First, the wave solution of Case III does not exist in the real frequency space since this type of solution represents a situation in which none of the wave components can propagate along the waveguide. Therefore the study of Case III is excluded in the present paper. Second, the vibrating motion of the shaft model in Case I is predominately pure shear [17] which 1296 is unique for the Timoshenko shaft model, while in Case U and Case IV the flexural mode and the simple shearing mode, which are corrected by including the rotary inertia and shearing effects in the formulation, dominate. Third, when the shaft rotates at a very high speed and/or the shaft is axially strained by tensile loads, the wave solution of Case IV governs the vibrating motion of the shaft model in the low frequency range. For comparison, the parameters A and B in the simple Euler-Bernoulli beam model are ( 10 a) ( 10 b) where, the non-dimensionalized natural frequency 6 ) is defined as is known as the bar velocity). ( 10 c) Note that, because B is negative, wave solutions of Case I and Case III do not exist. In general the displacement and the rotation of an infinitesimal shaft element consist of four wave components as shown by Eqns. (5a-8b). Once the displacement and the bending slope are known, the moment M and shear force V at a cross section can be determined from M = EI^, dz (ll) (du ^ y =J. (12) Moreover, the kinematic relationship between the transverse displacement and the slope due to bending is S^u 3^u . ,3 r 3r~ 3z^ 3z ' (I3a) where e' denotes the effects of the axial force and is defined as e = l + e —. a (I3b) 1297 3. WAVE REFLECTION AND TRANSMISSION AT SUPPORTS When a wave is incident upon a discontinuity, it is transmitted and reflected at different rates depending on the properties of the discontinuity. Consider a rotating Timoshenko shaft model supported at ^ = 0; see Fig. 2. The support simulates a bearing modelled by linear, translational and rotational springs, dampers, and a rotor mass which typically represents a gear transmitting a torque. Based on Eqns. (5a-8b), group the four wave components into 2x1 vectors of positive¬ travelling waves and negative-travelling waves C", i. e.. Recall that, depending on the system parameters, the rotating Timoshenko shaft model has four (practically three) different wave solutions in the entire frequency region as described in Eqns. (5a-8b). Thus C^ and C, in the above expression do not always correspond to propagating and attenuating wave components, respectively. When a set of positive-travelling waves is incident upon the support, it gives rise to a set of reflected waves C~ and transmitted waves . These waves are related by C'=rC" (15) D" = tC", (16) where r and t are the 2x2 reflection and transmission matrices respectively and are expressed as r = '^u ^ 12 * (17) / 2 ! '' 22 . ’^11 ^12 t = (18) / 2I ^22 _ From Eqns. (5a-8b), suppressing term and excluding Case III, the displacements u~ and and the bending slopes y/~and y/'^ at the left and right of z = 0, respectively, can be expressed in terms of the wave amplitudes of the displacement. For convenience, the over-bar (•) on the wavenumbers is dropped hereafter. Case 1 {A >0 andB>0); «-(z) = C,+ c;,£'^==, (19a) W'U) = -n,C:,e-‘^“ -77,c;,+T7,C>-'>''-(19b) 1298 M.Jm Figure 2. Wave motion at a general support (the disk may be considered as a gear transmitting a torque). where, _ 72-®' T/i — / » ~ / • r,£ 72^ // (A > 0 and B < 0): «-(z) = c;,+ C'e"''' + , i^-(z) = 7,,C,>-'"■= -77,C,-^r''''= +)7,C*e-''“ -TJ,C,;;e"=S k*(z) = A>“’'" + AV’‘. where = r,£' > Ba “ CaselV{A<0a.ndB<0): (19c) (19d) (20a, b) (21a) (21b) (21c) (21d) (22a, b) 1299 the following set of matrix equations can be established for each Case. Case / (A > 0 and 5 > 0): n n r 1 1 1 . ri n c^ + rC^=: L^i 12 ] -^2j -im, Ic+r IrC" KYi-tIi) i(r2“7?2)J [-KYi-Vi) -i(.Y 2 -Tl 2 )\ (28a) ri,(k,-J„,co^) + iTi,{c,0)~r,) 0)") +177272) ^28b) ik^~mO)-) + i{c,co + r^-r]^) (fc,-mtt>") + z’(c,® + 72- 772 ) J 1300 Case // (A > 0 and S < 0): 1 .^1 1 1 L^i 1 ^2 tc^ 1 1 n -ir,T7, [i(r,-T7i) r2-j772_ C -r -^r,-77,) ~ (r, - it ],)_ (29a) r},(k^-J„^0)-) + irij (c^co-r^) (k, -mco^) + i(c,Q) + r,-T],) Tl2(k,-J„,co^-r2) + iV2CrO} {kj -m(0^ +r2)+/(c,Ct}- 772) tc", (29b) Case /y(A<0and5<0); '1 11 ■ 1 1 ■ rC" = ' 1 r Jli Hi. -^2 -^ 1 . jii ^ 1 - ■-1X2772 -£t, 77 , ■ C'* -1X2772 KT.-ni) r,-i 77 ,_ -r — i(r 2 — 772) -(r, - in,). (30a) T]2ik^-J„^(0'") + iT]^{c,C0-y2) {k, -m(D‘) + i(c,aj + r2 - 772 ) riiik,-J„co^-y,) + iT],c^co' {k, -mco^ + r,) + i{c,(0-rij) tC", (30b) where Eqns. (15) and (16) have been applied in all Cases. Note that in Eqn. (27a), it is assumed that the rotational spring at the support is attached to the cross section of a shaft element such that the rotational spring responds only to the slope change due to rotation of the cross section and not the total slope change of the neutral axis of the shaft model. This assumption allows the shearing motion of the shaft element at the support. Note also that the effect of axial loads on the shear force at the support is neglected since the contribution of axial loads to the shear force at the support or boundary is small compared to the shear force due to the flexural motion of the shaft element. Exact moment and force balance conditions at boundaries for a rotating Timoshenko shaft element subjected to axial loads can be found in Ref. [16]. The corresponding matrix equations for the simple Euler-Bernoulli shaft model are shown in Appendix I. Solving the set of matrix equations simultaneously for r and t gives the elements of the reflection and transmission matrices for each Case. The general forms of solutions to these sets of equations for each Case is not presented in this paper due to space limitation. However one can obtain the solutions in either closed-form or numerically. Note that in Case II and Case IV, the first columns of r and t are the reflection and transmission coefficients due to incident propagating wave components, and the second columns are due to an incident attenuating wave component which is generally termed as near-field since this type of wave decays exponentially with distance. When the distance between the origin of disturbance and the discontinuity is very 1301 large, these attenuating wave components can be neglected. However, as mentioned by many authors, for example Graff [2], attenuating waves play an important role in wave motions by contributing a significant amount of energy to the propagating wave components when a set of propagating and attenuating waves are incident at a discontinuity and, in particular, when the distances between the discontinuities are relatively small, as in the case of closely-spaced multi¬ span beams. In this paper, near-field components are included. In what follows, the effects of the point supports on the reflection and transmission of an incident wave are studied. For comparison, the results are obtained for both the Timoshenko and the simple Euler-Bemoulli models, which hereafter, for brevity, are denoted by TM and EB, respectively. The system parameters used in the numerical results are taken from Ref. [10]; ao - 0.0955 m, p = 7700 kgW, K = 0.9, E = 207x10^ Wnf, G = 77.7x10^ N/ml 3.1. Wave reflection and transmission at rigid supports Consider two cases: the simple support and the clamped support. The r and t are solved and shown as follows. • Simple support (k, = oo, = m= c, = - 7„, =0) Case 7 (A > 0 and B > 0): r = _ 1 _ (72-7i)(r 1 / 2 + 6 ;') 7i(6>"-72) 72 ( 7 ?-6;-) 7i(6;’-72) (31a) t = _ 1 _ (72-7i)(7i72+6)-) 72 ( 7 ?-®') Case 77 (A > 0 and S < 0): _ 1 _ _ 1 _ (iT,-r,xr,r,-!V) Case 7V(A<0andB<0): _ 1 _ 7i(6)"“72) 7i(72-6)') r,(r,H®-) 1 r,{r^ + co^)' (31b) (32a) (32b) (33a) 1302 The corresponding reflection and transmission matrices for the EB model are listed in App. I Figures 3 and 4 plot the moduli (magnitudes) of the reflection and transmission coefficients for the simple and clamped supports. The finite cutoff frequencies, above which all waves propagate, are also marked in the figures. Thus, for the TM model, the wave motions change from Case 11 to Case 1 when (0>C0^ {0)^-4- in Fig. 3, co, = 4.24 in Fig. 4; O), is slightly altered by rotation speed and axial load). The results show that, at low frequencies (cD < 0.1 = 3156 rad/sec), the wave reflection and transmission coefficients of the TM model agree well with those of the EB model for both support conditions. However, as the frequency increases, the wave propagation characteristics of the TM model differ significantly from those of the EB model. These differences can be explained by examining the different modes of vibration. When co>co^ (in the regime of Case 1), the vibrating motion of the TM model is dominated by the pure shearing motion [7, 17], and hence the EB model, which neglects the rotary inertia and pure shear effects, become inaccurate at high frequency. As discussed in Ref. [7], at the finite cutoff frequency, the TM shaft experiences no transverse displacement, and the cross-section of the shaft simply rotates back and forth in unison In Figs. 3(d)-(f) and 4(d)-(f), for = 0 and £ = 0, the reflection and transmission coefficients of the EB model are independent of the frequency. This is because from Eqn. (10a), A = 0, and Eqns. (9c,d) lead to a single wavenumber F, = Tj. From Appendix I, Eqns. (32*-36*), the r and t are thus constant matrices. It is also seen that the wave reflection and transmission coefficients 1303 for both shaft models are basically independent of the rotation speed over the entire frequency range, even at high rotation speed = 0.05 = 44,600 rpm. In Ref. [7], it is also found that has negligible effects on the system frequency spectrum, phase velocity and group velocity. On the other hand, the effects of the axial load are significant for both propagating and attenuating waves in the regime of Case H, see Figs. 3(b)-(c) and 4(b)-(c). For both shaft models under simple support and compressive loads (Figs. 3(b, e)), the reflection coefficient ru of the incident propagating wave is reduced significantly in the regime of Case II, while the transmission coefficient fj, of the propagating wave component increases to balance the energy carried in the wave. However, the attenuating wave component which does not carry any energy loses its transmissibility in the same amount as the reflection coefficient r, 2 . Thus, in the presence of a compressive load, most of the transmitted wave energy in Case II comes from the propagating component of the incident wave. Note that axial tensile loads have the reverse effects on these wave components. In the clamped support case, the positive propagating wave component rn is constant over the regime of Case II under any loading conditions for the both shaft models, as seen in Fig. 4. Since there is no damping at the support, the incident power (Ilinc), reflected power (Hrefi) and transmitted power (Iltnin) in Cases II and IV are related by flinc = rirefl+ritran = Hinc, or Iriil^+Uiil^ = 1. This relationship is confirmed by the plots shown in Figs. 3 and 4, where for both shaft models, Irni and knl cannot exceed one. However in the regime of Case I, in which all wave components propagate, the energy balance is Hinc = (l^ii+r 2 iP+lfii+r 2 !p) Ilinc, or Hinc = (Iri 2 +r 22 l^+l?i 2 +r 22 l^) Hinc- Together with the plots on the phase of these coefficients (not shown to minimize the size of this manuscript), the above relationships can also be verified for wave motion of Case I. 1304 support without “resistance”. The impedance mismatching (rn = 1, ?,, = 0) frequency at which the propagating wave component is completely reflected without being transmitted can also be determined from Figs. 4(b, d) for the two shaft models. This impedance mismatching frequency is located in the regime of Case II for the TM model where the transverse mode dominates the vibrating motion of the shaft. Numerical results show that, as the spring constant increases, this impedance mismatching frequency increases, but is limited to within the regime of Case II and can never be found in the regime of Case I where the pure shearing mode dominates the vibrating motion of the shaft (refer to Fig. 3 for the transition of types of wave motion). Figure 6 shows the reflection and transmission coefficients for waves incident upon a support having both translational and rotational constraints. Since both flexural and shearing modes of vibration are constrained at this support, the maximum of the reflection coefficient is expected to be higher than the previous case. Figures 6(a-b) and (c-d) are the results for the TM and EB models, respectively. The translational and rotational spring constants used in the simulations are k,Q- 10^ N/m and ^^ 5 = 10^ Nm/rad, respectively. It is noted that in the regime of Case II, i.e.. Figure 6. Wave reflection and transmission coefficients at an elastic support with translational and rotational springs {k,-k,Q, k^^O, c, = c, = m= y,„ = 0) as a function of frequency, J3 = 0.05 and e = 0. (a-b) and (c-d) are results for the Timoshenko and Euler-Bernoulli shaft models, respectively. modulus modulus 2.0 r,2 (upper line) r,2=:r2, (middle dashed line) Tj, (lower line) Figure 4. Wave reflection coefficients at a clamped support (/c, = fc, = ■» and = c, = m = = 0) as a function of frequency, (a)-(c) and (d)-(f) are the results for the Timoshenko and Euler-BernouUi shaft models, respectively. The transition from one type of ivave motion to another is marked for the case /J = 0.05, £ = -0.05. 3.2, Wave reflection and transmission at elastic supports Figure 5 shows the reflection and transmission coefficients for waves incident upon a support with a finite translational spring for three different spring constants. Figures 5(a)-(b) and (c)-(d) are results for the TM and EB models, respectively. The spring constant used, k^Q = 10^ N/m, is a typical bearing spring constant value for turbine generators. The plots show that there is no significant difference in the moduli between the two shaft models. This is because the incident wave does not experience any rotational constraint at the support, and hence the additional rotary inertia factor in the TM model has only a small contribution to the wave motions. As the support spring constant increases, the curves for both the reflection and transmission coefficients are shifted to the right and, as the spring constant approaches infinity, these curves eventually become asymptotic to those shown in Fig. 3. Note that an impedance matching (r = 0, t = I), where all wave components are transmitted without being reflected, is found in the high frequency region for both shaft models. Thus, as the frequency increases, the characteristics of waves travelling along the shaft remain unchanged such that waves propagate through this elastic Figure 5. Wave reflection and transmission coefficients at an elastic support with a translational spring = c, = c, =m = =0) as a function of frequency, /3 = 0.05 and £ = 0. (a-b) and (c-d) are results for the Timoshenko and Euler-Bernoulli shaft models, respectively. 1308 in the low frequency range, both shaft models have similar reflection characteristics, and both the reflection and transmission coefficients are not significantly affected by the rotational spring. However, as the frequency increases, the effect of the rotational constraint on the wave motion becomes eminent, particularly for the TM model. As seen in Figs. 6(a) and 6(c), the reflection of the attenuating wave components are significantly higher than those of the propagating wave components. Hence, when a rotating shaft has a clamped support(s) such as a journal bearing, contributions from the attenuating wave components should be included in the formulation since a significant amount of energy in the propagating component arises from the incident attenuating wave component. It is noted that the impedance matching regions seen in Figs 5(a, c) disappear when the rotational constraint is added. Moreover, the impedance mismatching frequency shown in Figs. 5(b, d), which is found in the regime of Case II, also does not occur. At low frequency in Figs. 6 (b, d), there appears to be a mismatching region, but rj, is not exactly equal to zero. From Figs. 6(a, c), it is seen that there is a frequency at which the positive propagating wave component r,, is zero (this frequency is slightly different for the two models). This frequency does not correspond to an impedance matching, though the propagating wave is not reflected at all but is only transmitted (r,i = 1). Based on other research results [18], this phenomenon likely indicates a structural mode delocalization in bi-coupled systems, in which vibrations on both sides of the support become strongly coupled. Further research on the vibrations of rotating shafts with intermediate supports is being pursued to confirm the mode delocalization. Figure 7 plots the effects of axial compressive loads on the wave reflection and transmission upon a support with finite spring constant for the Timoshenko shaft model. As seen in Fig. 7(a), the reflection coefficient for the incident propagating wave component ru is substantially reduced in the low frequency range while the reflection coefficient for the incident attenuating wave component increases significantly. However, Fig. 7(b) shows the reversed effects on the transmission coefficient. It can therefore be concluded that, when the shaft is axially strained by Figure 7. Wave reflection and transmission coefficients at an elastic support (A:^ = k,Q and kr = c, = Cr = m = J„, = 0) for the Timoshenko shaft model with and without the compressive load, (a) reflection coefficients, (b) transmission coefficients. 1309 Figure 8. Wave reflection and transmission coefficients at an elastic support k^ = and c,= Cr = m = J„, = 0) for the Timoshenko shaft model with and without the compressive load, (a) reflection coefficients, (b) transmission coefficients. compressive loads, the energy contribution from the incident attenuating wave component to the energy in the reflected propagating wave is more significant than the strain-free situation in the low frequency range, while most of the energy in the transmitted wave derives from the incident propagating wave component. Figure 8 plots the wave reflection and transmission coefficients along an axially compressed Timoshenko shaft model at a support with finite translational and rotational spring constants. Similar results to the previous example can be observed in terms of energy contribution from the incident attenuating wave component in the low frequency range. However, the effects of the axial compressive load on both the reflection and transmission coefficients for the propagating wave component ( r^^ and r,,) are significantly reduced when compared to Fig. 7. 3.3. Wave reflection and transmission at damped supports Figure 9 shows the effects of both translational and rotational dampers at a support with finite translational and rotational spring constants. Figures 9(a, b) and (c, d) are results for the TM and EB models, respectively. The translational and rotational damping constants used in this study are c^to = 2x10^ Ns/m and Cdro = 64x10^ N-m-s/rad, typical values for bearings in turbine generators. The curves with symbols (• and ♦) are the results when the rotational damping factor is also included in the formulation. It can be seen that I and Ir^l for both shaft models are significantly lowered due to the presence of damping. Note that, because of the damping, the frequency at which 1 rj, I = 0 (compare with Figs. 6(a, c)) no longer exists for both shaft models. It can also be seen that the effect of the rotational damping factor on the wave reflection and transmission is not significant over the entire frequency range for both shaft models. For TM model, the contribution of the rotational damping to both Ir;, 1 and 1?,, I is almost negligible. The 1310 support condition considered in this particular example is simulated as an actual bearing support adopted in turbine generators. Hence for this particular type of bearing support, the effect of the rotational damping on wave reflection and transmission is not considerable. Other numerical results (not shown in this paper) show that the wave propagation at the damped support is characterized by translational damping rather than rotational damping. Note that similar results have been presented for the support without damping (see Fig. 6). Figure 9. Wave reflection and transmission coefficients at an elastic support with damping (k, = k,o, K = Ko, c, = c,;,o, c, = cjro and m = J,„ = 0) as a function of frequency for /3 = 0.05 and £ = 0. (a-b) and (c-d) are results for the Timoshenko and Euler-Bernoulli shaft models, respectively. 3.4. Wave reflection and transmission at a rotor mass Consider a gear rigidly assembled to a rotating shaft. The gear is assumed to be perfectly balanced and its thickness is sufficiently small such that wave reflection and transmission due to the geometric discontinuity between the shaft and the gear can be neglected. However the gear does resist the translational and rotational motions of the cross-sectional element of the shaft. 1311 Figure 10 shows the reflection and transmission upon the gear when the mass mo and mass moment of inertia J^o of the gear are 4 and 16 times of the shaft, respectively. Not shown in Figs. 10 (b, d) is that r ,2 = 0 when w= /„ = 0. Like some previous support conditions discussed, the effects of the rotor mass are much more significant in the high frequency region for both models (particularly around and beyond the cutoff frequency for the TM model). In general, the rotor mass decreases the transmission and increases the reflection of the wave. At very high frequency, there is basically no wave transmission. Note that, since the geometric discontinuity between the shaft and the gear is neglected in this model, one may expect that the actual reflection for both the propagating and attenuating wave components would be higher. (a) (m= = V (b) -(m = 0, -( m = mg. ) / Figure 10. Wave reflection and transmission at a rotor mass assembled to a rotating shaft (*, = 0 = L = Cf = Cr = 0, and m = rriQ, and J,„ = J„^) as a function of frequency when P = 0.05 and £ = 0. (a-b) and (c-d) are results for the Timoshenko and Euler-Bernoulli shaft models, respectively. 4. WAVE REFLECTION AND TRANSMISSION AT A GEOMETRIC DISCONTINUITY It is common for a rotating shaft element to have changes in cross-section, or to be joined to 1312 Case I, II, or IV A, = (1 + a)co- - 2j3co - 16£ (1 + £ - ~) a Z=0 Case I, II, or IV B, = co' ccco- - 2p(o - I6a (1 + £)(1 + £-) a A={\ + a)(o^-2^co-^^^ (l + £, -—) o" a 16a = co^\a (O' -215(0-^^ (l+£j(l + £ -— ) L O’ a Figure 11. Wave reflection and transmission at a geometric discontinuity. another shaft element by a coupling. Figure 11 shows a typical example of a discontinuous shaft model in which two shafts of differing wavenumber and diameter are joined at z = 0. The subscripts I and r denote z = O' and z = 0^ regions, respectively. It is known that when a wave encounters a junction or a discontinuity, its wavenumber is changed. It is therefore possible that a wave on the left side of the junction can be propagating, while after crossing the junction to the right side, the wave becomes attenuating. Therefore, for a Timoshenko shaft, when a wave propagates through the junction, there are mathematically nine possible different combinations of wave motions to be considered depending on the values of the functions A and B on each side of the junction, as depicted in Fig. 12. Figure 12. Nine possible combinations of wave motions at a geometric discontinuity of the cross section for the Timoshenko shaft model. Subscripts / and r denote the left and the right side of the discontinuity, respectively. 1313 For simplicity, assume that material properties such as p, E, and G are the same for both sides of shaft element. The displacement continuity, moment and force equilibrium conditions are applied at the junction to determine the wave refection and transmission matrices. Results for the three most commonly encountered possibilities in the low frequency regime are listed as follows. Case II {A, >0, Bi <0) - Case I (A^ > 0, R, >0): ■ 1 r 1 1 ' r 1 M rC" = tCY Jlu ^21. nu ■ ~n2i. Jl\r ^2r_ r -i^Bu ^21^2! C" + r Xtnu ”Y,r72, 1 L^xr., -riu) r 21-in 21^ 1 Y rr 1 T ^772,)J rC-^ Yuriu 72rn2r L-io-"(r„-77„) -i<y^(r2r-V2r)J tc", Case II (A, >0, B, <0) - Case II (A^ > 0, B^<0): ■ 1 1 ■ ■ 1 1 ■ ■ 1 1 * C" + rC^ = Jlv n2l. — nu “^2/. Y.r n2r. (37a) (37b) (38a) -"^21^21 ^2, -in21 C" + i^unu ^21^21 ~i(^H — nu ) ~(^2/ ~ ^^21 ) rC" -i<yXrnu -^Xr'n2r -zcr"(r,,-77„) -C7“(r2,-7]2,)J CaseII{A^>0, B, <0) - Case IV {A, <0, S, <0): r i 1 1 r 1 11 ■ 1 r C" + rC^ = nu n 2 i l-nu - n 2 i ^ n 2 r n ^ r . Y/^2/ ^2/^2/ JXi—nu) Y/~^*^2;. “(Yf -in2i)_ -i^%rn2r -<yXrn:r tc\ (38b) (39a) (39b) where cr is the diameter ratio between the shaft elements, defined as 1314 ( 40 ) CJ = —. Note that 77/5 in Eqns. (37a-39b) are given by Eqns. (20a, b), (22a, b), and (24a, b) according to the type of wave motion, and 77 / s on the right side of the geometric discontinuity are modified as follows. where : - 77 =-— for Case I, r„e; yzX (41a, b) -co^ Tl+co- , Ti, - — - 7 - for Case II, r„e; iT^rS; (42a, b) ,7]2 for Case IV, ir,x r,,e; (43a, b) £' = 1 + 8,. —and £, = — . (44) a O' Moreover, the wavenumbers, A and B of the shaft element on the right side of the junction are modified as follows. y„ = ^[a, + ^Aj-4B^f . r,, = - Va?- 4B,)^ , (45a. b) + = . (45c,d) where, A. = (l + a)a.= -2pa)-^a + £,-|), B. =co‘ am--2Pa-^ (l + £,)(l + £,-^) C OL (46b) Corresponding results for the simple Euler-Bemoulli shaft model are listed in Appendix II. Figures 13 to 16 show some representative examples of wave reflection and transmission upon the geometric discontinuity. In Figs. 13 and 14, the thick and thin curves represent results 1315 for the TM and EB models, respectively. The second graph in each figure shows the changes of Ai, Bi, Ar and Br, and how wave solutions on both sides of the discontinuity change as the frequency increases for the TM model. In general, the wave reflection and transmission for the EB model are frequency independent except when the shaft is axially strained, while the wave propagation characteristics for the TM model are strongly dependent on the frequency. Comparing Figs. 13 and 14, it is noted that, for both shaft models, the average reflection and transmission rates for cr = 0.8 are higher than those for o' = 1.2, especially for the attenuating wave components. These results imply that incident attenuating waves contribute more energy to propagating waves at the discontinuity when the waves travel from a smaller to a larger cross- section. In particular, it is noted that the transmissibiiity of the attenuating wave tn has a strong dependency on the direction of propagation. Note also that the differences between the two shaft models are more pronounced when cr = 0.8. It is clearly seen from the figures that when Bj and B^ change from negative to positive, both reflection and transmission coefficients experience a sharp jump or drop at the finite cutoff frequencies, due to changes in the types of wave motion. In the frequency region (S, > 0 and B^ <0) located between the two cutoff frequencies in Fig. 2.0 h 0.5 Thick curves : Timoshenko shaft model Thin curves : Euier-Bernoulli shaft model - Ui h i il _ - 7/ X.-V' .'7 \ 5 (O Figure 13. Reflection and transmission of waves incident upon a change in the cross-section, a= 0.8, P = 0.05, and £ = 0. Thick and thin curves are results for the Timoshenko and Euier-Bernoulli shaft models, respectively. Note that the ordinates in the lower graphs keep increasing with frequency (abscissa). 1316 13, the wave motion on the left side of the junction is governed by the wave solution of Case I since all wave components are propagating at a frequency larger than the cutoff frequency, while the wave motion on the right side of the junction is governed by the wave solution of Case II. Thus, for O’ = 0.8, some of the propagating wave components on the left side of the shaft element cannot propagate as they pass the discontinuity, and become attenuating. A similar, but converse conclusion can be drawn for the frequency region (5^ > 0, 5, < 0) when cr = 1.2, as shown in Fig. 14. The results of Figs. 13 and 14 show that, for different system parameters cr, jS, and e and at any given frequency, the types of wave motion on each side of the discontinuity can be different, as depicted in Fig. 12. From Eqns. (41a-43b), it is seen that when the Timoshenko shaft is axially strained and (O is not sufficiently large, the wavenumber (hence wave propagation characteristics) depends strongly on the cross-section ratio <7. Figure 15 shows the effects of the axial load on the wave reflection and transmission, which are mostly limited to the relatively low frequency region. In Figs. 15(a- b), when the shaft is axially compressed (£ = -0.05), the reflection and transmission due to the incident attenuating wave component decrease for both o < 1 (plot (a)) and <7 > 1 (plot (b)). However, the transmission due to an incident propagating wave decreases significantly for — — f, 2 2.0 h Figure 14. Reflection and transmission of waves incident upon a change in the cross-section, cr= 1.2, /3 = 0.05, and £ = 0. Thick and thin curves are results for the Timoshenko and Euler-Bernoulli shaft models, respectively. Note that the ordinates in the lower graphs keep increasing with frequency (abscissa). 1317 Figure 15. Reflection and transmission of waves upon a change in the cross-section when = 0.05 for the Timoshenko shaft model, (a) < 7 = 0.8 and £ = -0.05. (b) (T= 1.2 and e=-0.05. (c) cr= 0.8 and £= 0.05. (d) or = 1.2 and e= 0.05. Thin and thick curves show the results when the shaft is strain-free (£= 0) and strained, respectively. c - 0.8 and increases for <7 = 1.2 at low frequency. Effects of the axial load on the wave reflection and transmission are more significant when the shaft is compressed (Figs. 15(a-b)) than when it is under tension (Figs. 15(c-d)). This is because the wavenumbers of both the propagating and attenuating wave components are only slightly changed. It is also noted that, in the low frequency range, the wave solution of Case IV governs the wave motions on both sides of the discontinuity, and the wave components which have large wavenumber (Fi) attenuate, while wave components with small wavenumber (r 2 ) propagate along the waveguide as long as A remains negative. 1318 5. WAVE REFLECTION AT BOUNDARIES When a wave is incident upon a boundary, it is only reflected because no waveguide exists beyond the boundary. Consider an arbitrary boundary condition with translational and rotational spring constraints, dampers, and a rotor mass, as shown in Fig. 16. The reflection matrix at the boundary is derived for each Case. Applying the same non-dimensional parameters employed in Section 3, and by imposing the force and moment balances at the boundary, which can be deduced by eliminating and inEqns. (27a, b), M~ = k^y/ + c^yr + J„y/, (47 a) -V = k,u + c,u + mu, (47b) the reflection matrix for each Case is determined. CaseI{A>0, B>0): J r72(ir2-2:,J T’r-T7,(jri + ^.) + + Kr2-Bi) + ^s\ Kr2-T72)-^.J’ Case 7/ (A > 0, 5 < 0): 7],(ir,-2:j T'r-77,(ir,+Ej -7]2(r2 + zj1 '''[i(r,-r7,)+z, (r,-ii7,)+i.J [i(r,-j?,)-z, M,Jm Z=0 Figure 16. Wave reflection upon a general boundary. 1319 CaseIV{A<0, 5<0): 772 (^r2 s„,) iCFj - 7 ) 2 ) + (r.-mj+s. 772 (zTj + Sp,) /(r2 ~ ^ 2 ) ■“ -^iCr.+zj (r,-j77,)-E, (50) where 77 's in above equations have been defined in Eqns. (20a, b), (22a, b) and (24a, b), and Z„, = + ic^co - J,„co ~, and = ^, + ic,Q) - mco^ , (51a, b) The corresponding results for the simple Euler-Bernoulli shaft model are listed in the Appendix in. By specifying the parameters in the reflection matrix r, results for three typical boundary conditions (simple support, clamped support, and free end) can be obtained. • Simple support {k, =^, =m = c, = = 7,„ = 0) r -1 0 0 -1 for Case I, II, and IV, (52) • Clamped support (k^ = k,. = m = c, = = J„, = 0) 1 Tli-ri2 (53b) (53c) • Free end {k, = k^ = m = c, = = /„, = 0, and £ = 0) 1 rii+ri2 2772 77,- 772 L-277i -(r?,+772)J 1 -irii+Tli) "2772 2t}, ( 771 + 772 ) for Case II, for Case IV, ■ 77 ,+7?2 2772 - 277 , -( 771 + 772 ), for Case I, (53a) r J_ 77,7?2(ri+72)”7ir2(77,+ 772 ) -277,7i( 77,-7i) 277272 ( 772 - 72 ) - 77 , 772 ( 71 +72) + 7i72(77i+ 772 ). (54a) where. A, = r],n, - 7, ) + 7,72 (77, - ^2 ) for Case I, + 72 ) + 7 i 72 ( 77 i + 772 ) 2i7}^y^iin^-y^) ^ A;, [ 2/77,7,(771-7,) 77,772(/7,+72)-7 i 72(7?,+772) (54b) where, = 77 , 772 ( 77 , - 72 )- 7 , 72 ( 7 ?, - 772 ) for Case//, 1320 Figure 17. An example of a rotating shaft with multiple supports and discontinuities. I \TiiV 2 (ri+ir 2 )-rj 2 (Tii+r] 2 ) 277,7,(77,+ir,) ' ^ r =- (54c) A/v L -2i7?2r,(77j-7j - 77 , 772 ( 7 , +J72)+rir2(^i +^ 2 ). where, A;^ = 77 , 772(71 - 172 ) “7172(Hi- 772 ) for Case IV. 6. APPLICATIONS The reflection and transmission matrices for waves incident upon a general point support or a change in cross-section can be combined with the transfer matrix method to analyze the free vibration of a rotating Timoshenko shaft with multiple supports and discontinuities, and general boundary conditions. The basic idea of this technique has been shown in Ref. [6]. However, due to the complex wave motions in the Timoshenko shaft model, such as the frequency dependency of the wave reflection and transmission at a cross-section change, it is important to apply the proper reflection and transmission matrices consistent with the values of A and B on both sides of the discontinuity, particularly when numerical calculations are performed. Consider for example the free vibration problem of the rotating Timoshenko shaft model shown in Fig. 17. Denoting R as a reflection matrix which relates the amplitudes of negative and positive travelling waves at a discontinuity, and defining T/ as the field transfer matrix which relates the wave amplitudes by C^iZo + z) = TC"(Zo), C-(zo + z) = T-'C-(zo), (55) the following relations can be found. 1321 W-=R5W^ (R5=>-s). (56a) i = 2,3,4 (station number), /i = left (/) or right (r) (56b) w>T,w-,, (56c) < =r,w-, (56d) <=T.w;, (56e) where in Eqn. (56b), R,,=TiR„,,T,. R,,=r,+t,(R-’-r,-)-'t, (56f) Solving the above matrix equations gives (r,T,R„T,-I)w>0, (57) where each element of the matrix is a function of two different wavenumbers and the frequency CO. For non-trivial solutions, the natural frequencies are obtained from the characteristic equation Det[(r,T,R„T,-I)] = 0. (58) The proposed method is applied to an example of a two-span rotating shaft, simply supported at the ends and with an intermediate support consisting of translational and rotational springs, as shown in Fig. 18. Numerical computations were performed by a PC-based Mathematical. The values of the spring constants Kj and Kr are those introduced in Section 3.2, with ^ = 1 m and the rotation parameter p = 0.05. 1322 Figure 19 shows the first eight natural frequencies of the vibrating shaft for both the TM and EB models for a classical simple intermediate fixed support K^=0) placed at various locations. The results confirm the well-known fact that the Timoshenko model leads to smaller eigenvalues. Figure 20 shows the first eight natural frequencies of the Timoshenko shaft for an elastic intermediate support with three different translational spring constants. It can be seen that the effect of the translational spring diminishes for higher modes. The proposed wave analysis technique can also be applied effectively to the study of structural mode localizations in mistuned, rotating systems. Dynamics of such systems will be addressed in another paper. Figure 19. Natural frequencies of a two-span, rotating shaft as a function of the support location; intermediate support is fixed. Figure 20. Natural frequencies of a two-span rotating Timoshenko shaft as a function of the support location; intermediate support consists of k, and kr. 7. SUMMARY AND CONCLUSIONS In modern high speed rotating shaft applications, it is common that the shaft has multiple intermediate supports and discontinuities such as bearings, rotor masses, and changes in cross- 1323 sections. In many cases, the ratio of the shaft diameter to its length between consecutive supports is large, and the Timoshenko model (TM) is needed to accurately account for the shear and rotary inertia effects. In this paper, the wave propagation in a rotating, axially strained Timoshenko shaft model with multiple discontinuities is examined. The effect of the static axial deformation due to an axial load is also included in the model. Based on results from Ref. [7], there are four possible types of wave motions {Cases I, II, III and IV) in the Timoshenko shaft, as shown by Eqns. (5a-8b). In practice. Case III does not occur and is excluded in the analysis. For each Case, the wave reflection and transmission matrices are derived for a shaft under various support and boundary conditions. Results are compared with those obtained by using the simple Euler- Bemoulli model (EB) and are summarized as follows. 1) In general, the two shaft models show good agreement in the low frequency range where the wave motion is governed by Case II and Case IV. However, at high frequencies, the types of wave motions and propagation characteristics for the TM and EB models are very different. 2) The effects of shaft rotation on the wave reflection and transmission are negligible over the entire frequency range and even at high speed (up to 44,600 rpm). While the effects of the axial load are significant, especially in the low frequency range. 3) When waves are incident at supports with only translational springs, differences in the results between the TM and EB models are small, and there exists frequency regions of impedance matching and an impedance mismatching frequency (limited to within the regime of Case IT). The impedance matching and mismatching disappear when a rotational spring is added to the support. Instead, there is a frequency at which Irul = 0 and Ifni = 1, and vibrations on both sides of the support become strongly coupled. This (delocalization) phenomenon suggests further research on the vibrations of constrained multi-span beams. When there is damping at the support, the frequency at which IrnI = 0 does not occur. Moreover, effects of translational damping on the wave propagation are more significant at high frequency, especially for the TM model, however effects of rotational damping is not significant over the entire frequency range. 4) Contributions of attenuating wave components to the energy in the reflected and transmitted waves are significant when the shaft is axially strained and when the support has a rotational constraint. Thus attenuating waves should be included in the formulation. 5) Unlike the spring supports, in which waves are easily transmitted at high frequency, the rotor mass support diminishes the wave transmission as the frequency increases. 6) When waves are incident at a geometric discontinuity such as a change in the cross-section, there are nine possible combinations of wave motions on both sides of the discontinuity. It is shown that differences of the results between the TM and EB models depend on the diameter ratio (and hence the direction of the wave incidence). Moreover, incident attenuating waves contribute more energy to propagating waves at the discontinuity when the waves travel from a smaller to a larger cross-section. When the shaft is axially strained, the effects of the load on the wave propagation are primarily limited to the low frequency range. The reflection and transmission matrices are combined with the transfer matrix method to provide a systematic solution method to analyze the free vibration of a multi-span, rotating shaft. Since the procedure involves only 2x2 matrices (while including the near-field effects already), strenuous computations associated with large-order matrices are eliminated. 1324 ACKNOWLEDGMENTS The authors wish to acknowledge the support of the National Science Foundation and the Institute of Manufacturing Research of Wayne State University for this research work. REFERENCES 1. Lin, Y.K., Free Vibrations of a Continuous Beam on Elastic Supports. International Journal of Mechanical Sciences, 1962, 4, pp. 409-423. 2. Graff, K.F., Wave Motion in Elastic Solids, Ohio State University Press, 1975. 3. Cremer, L,, Heckl, M. and Ungar E.E., Structure-Bome Sound, Springer-Verlag, Berlin, 1973. 4. Fahy, F., Sound and Structural Vibration, Academic Press, 1985. 5. Mead, D.J., Waves and Modes in Finite Beams: Application of the Phase-Closure Principle. Journal of Sound and Vibration, 1994, 171 , pp. 695-702. 6. Mace, B.R., Wave Reflection and Transmission in Beams. Journal of Sound and Vibration, 1984, 97 , pp. 237-246. 7. Kang, B. and Tan, C.A., Elastic Wave Motions in an Axially Strained, Infinitely Long Rotating Timoshenko Shaft. Journal of Sound and Vibration (submitted), 1997. 8. Han, R.P.S. and Zu, J.W.-Z., Modal Analysis of Rotating Shafts: A Body-Fixed Axis Formulation Approach. Journal of Sound and Vibration, 1992,156, pp. 1-16. 9. Zu, J.W.-Z. and Han, R.P.S., Natural Frequencies and Normal Modes of a Spinning Timoshenko Beam With General Boundary Conditions. Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics, 1992, 59, pp. 197-204. 10. Katz, R., Lee, C.W., Ulsoy, A.G. and Scott, R.A., The Dynamic Response of a Rotating Shaft Subject to a Moving Load. Journal of Sound and Vibration, 1988,122, pp. 131-148. 11. Tan, C.A. and Kuang, W., Vibration of a Rotating Discontinuous Shaft by the Distributed Transfer Function Method. Journal of Sound and Vibration, 1995,183, pp. 451-474. 12. Argento, A. and Scott, R.A., Elastic Wave Propagation in a Timoshenko Beam Spinning about Its Longitudinal Axis. Wave Motion, 1995, 21 , pp. 67-74. 13. Dimentberg, F.M., Flexural Vibrations of Rotating Shafts, Butterworth, London, 1961. 14. Dimarogonas, A.D. and Paipeties, S.A., Analytical Method in Rotor Dynamics, Applied Science, New York, 1983. 15. Lee, C.W., Vibration Analysis of Rotors, Kluwer Academic Publishers, 1993. 16. Choi, S.H., Pierre, C. and Ulsoy, A.G., Consistent Modeling of Rotating Timoshenko Shafts Subject to Axial Loads. Journal of Vibration and Acoustics, 1992, 114 , pp. 249-259. 17. Bhashyam, G.R. and Prathap, G., The Second Frequency Spectrum of Timoshenko Beams. 1325 Journal of Sound and Vibration, 1981,76, pp. 407-420. 18. Riedel, C.H. and Tan, C. A., Mode Localization and Delocalization of Constrained Strings and Beams. Proceedings ofASME Biennial Conference on Mechanical Vibration and Noise (submitted), 1997 1326 For simple and clamped supports, the reflection and transmission matrices are listed as follows. Simple support (k. = k. = n l = C, = c, = . Case // (A > 0, B < 0): 1 ■ r. r, ■ iF, r2 'ir, r./ '"ir.-r2 .-ir^ -r^. CaseIV{A<0, B<0): 1 -ire 1 1 t “ 'r. -ir: iT, +r2 .-^2 -n_ i ^T,+r2 -r2 • Clamped support {k^ =<=<>, m=c, = c, = J„, = 0); t = 0. CaseII{A>0, B<0): 1 ^^ 1+^2 2 r 2 -2iT, -(ir, + r,)J’ CaseIV{A<0, B<0): 1 r-(ir,-r2) -2iTi ' ^“iT.+r.L -2r2 iT,-r^ ■ (32a*, b*) (33a*, b*) (35*) (36*) APPENDIX II The reflection and transmission matrices for a wave incident upon a cross-sectional change for the simple Euler-Bernoulli shaft model can be determined by solving the following sets of matrix equations. Only two representative combinations are shown. CaseII(Ai>0, B, <0)-Case II {A,>0, B,<0): 1327 Casen{A,>0, <Q) - Case TV {A, <0, <0): 1 1 1 11 . 1 1 1 . ■ + - •^2,. -F^ F" ^1/ •*■2/ C* + F2 1 rC" = [T^ _r3 L “ 1/ ^ 21 2/J r,rj where, F sr and T 2 r have been defined in Eqns. (45c, d), and A^- and Br are given by O’" G (39a') (39b') (46a') (46b*) If the rotating shaft is strain-free, then r can be reduced to simple forms representing typical boundary conditions such as simple support, clamped support, and free end as shown in Ref. [6]. Note that for those supports in the strain-free case, the reflection matrices are constant. 1328 ANALYTICAL MODELLING OF COUPLED VIBRATIONS OF ELASTICALLY SUPPORTED CHANNELS Yavuz YAMAN Department of Aeronautical Engineering, Middle East Technical University 06531 Ankara, Turkey An exact analytical method is presented for the analysis of forced vibrations of uniform thickness, open-section channels which are elastically supported at their ends. The centroids and the shear centers of the channel cross-sections do not coincide; hence the flexural and the torsional vibrations are coupled. Ends of the channels are constrained with springs which provide finite transverse, rotational and torsional stiffnesses. During the analysis, excitation is taken in the form of a point harmonic force and the channels are assumed to be of type Euler-Bernoulli beam with St.Venant torsion and torsional warping stiffness. The study uses the wave propagation approach in constructing the analytical model. Both uncoupled and double coupling analyses are performed. Various response and mode shape curves are presented. 1. INTRODUCTION Open-section channels are widely used in aeronautical structures as stiffeners. These are usually made of beams in which the centroids of the cross-section and the shear centers do not coincide. This, inevitably leads to the coupling of possible flexural and torsional vibrations. If the channels are symmetric with respect to an axis, the flexural vibrations in one direction and the torsional vibrations are coupled. The flexural vibrations in mutually perpendicular direction occur independently. In the context of this study, this type of coupling is referred to as double-coupling. If there is no cross-sectional symmetry, all the flexural and torsional vibrations are coupled. This is called as triple-coupling. The coupling mechanism alters the otherwise uncoupled response characteristics of the structure to a great extent. This problem have intrigued the scientists for long time. Gere et al [1], Lin [2], Dokumaci [3] and Bishop et al [4] developed exact analytical models for the determination of coupled vibration characteristics. All those works, though pioneering in nature, basically aimed to determine the free vibration characteristics of open-section channels. The method proposed by Cremer et al [5] allowed the determination of forced vibration characteristics, provided that the structure is uniform in cross-section. The use of that method was found to be extremely useful when the responses of uniform structures to point harmonic forces or line harmonic loads were calculated. Mead and Yaman presented analytical models for the 1329 analysis of forced vibrations of Euler-Bernoulli beams [6]. In that they considered finite length beams , being periodic or non-periodic, and studied the effects of various classical or non-classical boundary conditions on the flexural response. Yaman in [7] developed mathematical models for the analysis of the infinite and periodic beams, periodic or non-periodic Kirchoff plates and three-layered, highly damped sandwich plates. Yaman in [8] also developed analytical models for the coupled vibration analysis of doubly and triply coupled channels having classical end boundary conditions. In that the coupled vibration characteristics are expressed in terms of the coupled wave numbers of the structures. The structures are first assumed to be infinite in length, and hence the displacements due to external forcing(s) are formulated. The displacements due to the waves reflected from the ends of the finite structure are also separately determined. Through the superposition of these two, a displacement field is proposed. The application of the end boundary conditions gives the unknowns of the model. The analytical method yields a matrix equation of unknowns which is to be solved numerically. The order of the matrix equation varies depending on the number of coupled waves. If the cross-section is symmetric with respect to an axis ( double-coupling) and if the warping constraint is neglected, the order is six. If there is no cross-sectional symmetry (triple-coupling) and if one also includes the effects of warping constraint, the order then becomes twelve. This order is independent of the number of externally applied point forces. Although the method is basically intended to calculate the forced response characteristics, it conveniently allows the computation of free vibration characteristics as well. The velocity or acceleration of a point can easily be found. The mode shapes can also be determined. Both undamped and damped analyses can be undertaken. This study is based on the models developed in reference [8] and aims to analyze the effects of non-classical end boundary conditions on the coupled vibratory responses. If the ends are elastically supported (which may also have inertial properties) the problem becomes so tedious to tackle through the means of classical analytical approaches. The current method alleviates the difficulties encountered in the consideration of complex end boundary conditions. In this study a typical channel, assumed to be of type Euler-Bernoulli beam, is analyzed. It represents the double-coupling. Effects of the elastic end boundary conditions on the resonance frequencies, response levels and mode shapes are analyzed. Characteristics of otherwise uncoupled vibrations are also shown. 1330 2. THEORY 2.1 Flexural Wave Propagation in Uniform Euler-Bernoulli Beams Consider a uniform Euler-Bernoulli beam of length L which is subjected to a harmonically varying point force Fo e acting at x=Xf. The total flexural displacement of the beam at any Xr (0 < Xr < L) can be found to be [5-8], w(x,,t) = ( i;A„e''„\ + Fo i ane'^' r r )e' ( 1 ) The first series of the equation represents the effects of four waves which are being reflected from the ends of the finite beam. They are called free-waves. The second series accounts for the waves which are being created by the application of the external force Fo e on the infinite beam. Those waves are known as forced-waves, kn is the n’th wave number of the beam and kn =(mco^/EI)'^'‘ where m= Mass per unit length of the beam, co= Angular frequency, EI= Flexural rigidity of the beam, an values are the complex coefficients which are to be found by satisfying the relevant compatibiliy and continuity conditions at the point of application of the harmonic force [6,7]. An values, on the other hand are the complex amplitudes of the free waves and are found by satisfying the required boundary conditions at the ends of the beam. Once determined, their substitution to equation (1) yields the flexural displacement at any point on the finite beam due to a transversely applied point harmonic force. More comprehensive information can be found in [7]. 2.2 Torsional Wave Propagation in Uniform Bars If one requires to determine the torsional displacements generated by a point, harmonically varying torque, a similar approach to the one given in Section 2.1 can be used. In that case, the total torsional displacement can be written as: = )e‘”' (2) k is the wave number of the purely torsional wave and is known to be k=(-pIoa)VGJ)^^^ . k 2 = -ki and GJ=Torsional rigidity of the beam, p=Material density, Io=Polar second moment of area of the cross-section with respect to the shear centre. Toe'“ ^ is the external harmonic torque applied at x=Xt and b=l/(2kGJ). Bn values are the complex amplitudes of the torsional free-waves and are found by satisfying the appropriate end torsional boundary conditions. The consideration of the warping constraint To modifies equation (2) to the following form. 1331 ( 3 ) «x„t) = (£C„e^\ + Toic„e-'=„'V,' ) e n*l »•! Now kn are the roots of EFo kn'^-GJkn^-pIoCO^=0 ( 4 ) Cn values are found by satisfying the necessary equilibrium and compatibility conditions at the point of application of the point harmonic torque acting on an infinite bar [8]. Cn values are determined from the end torsional boundary conditions of the finite bar. 2.3 End Boundary Conditions for Uncoupled Vibrations 2.3.1 Purely Flexural Vibrations Consider an Euler-Bernoulli beam of length L which is supported by springs at its ends. The springs provide finite transverse and rotational constraints Kt and Kr respectively. The elastic end boundary conditions can be foimd to be: El w”(0) - Kr I w’(0)=0 El w’”(0) + Ktj w(0)=0 El w”(L) + K r,r w’(L)=0 El w’”(L) - Kt,rW(L)=0 (5) Here w’= dw(x) w d‘w(x) and w’” dV(x) w(x) is the spatially dx ’ dx“ dx' dependent part of equation (1) and second subscripts 1 and r allows one to use different stiffnesses for left and right ends. A more comprehensive study on these aspects can be found in references [6,7]. 2.3.2 Purely Torsional Vibrations Now consider a bar of length L which is supported by torsional springs, having finite Ktor,at its ends. The elastic end boundary conditions requires that, Torque (0) - Ktor,i (|)(0)=0 and Torque (L) + Ktor .r (i)(L)=0 (6) Depending on the consideration of the warping constraint To, the torque has the following forms Torque(x)=GJ —— or Torque(x)= GJ —— - Ei o , 3 (/) dx dx ux 1332 (})(x) in equation (7) should be obtained either from equation (2) or equation (3) depending on the warping constraint r©. 2.4 Doubly-Coupled Vibrations Now, consider Figure 1 . It defines a typical open cross-section which is synunetric with respect to y axis V z (b) Figure 1 : A Typical Cross-section of Double-coupling ( a. Coordinate System, b. Real and Effective Loadings C: Centroid, O: Shear Centre) A transverse load applied through C results in a transverse load through O and a twisting torque about O. In this case the flexural vibrations in z direction are coupled with the torsional vibrations whereas the flexural vibrations in y direction occur independently. The motion equation of the coupled vibrations is known to be [1,2]. a-w a-(j) = 0 a‘‘(i) 3^6 a’w a"(j) ( 8 ) If one assumes that, w(x,t) = w„eVe”' (|)(x.t) = 4>„eVe“’ (9) 1333 Then, it can be found that, a load Pz through the centroid will create the following displacements at any x (0 < x < L) along the length of the channel [8], w(x,t) = (2A„e^’‘+Pjt a„ e'^ )e'"' n^l nvl n«l n>I Now kn values are the coupled wave numbers, An values are the complex amplitudes of the coupled free waves, an values are the complex coefficients which are to be found by satisfying the required compatibility and continuity conditions and 'Pn^ ( (El^ kn'^-mco^) / (CymcD^)) [8]. If required, the warping displacement u(x,t) can be found from (l)(x,t )as u(x,t)=-2A,^iM (11) dx where As is the swept area. Here 2j gives the order of the motion equation. j=3 defines the case in which the effects of warping constraint are neglected and j=4 represents the case which includes the warping effects. An values are found by satisfying the necessary 2j end boundary conditions. If warping constraint is neglected, the required six boundary conditions have the general forms given in equations (5) and (6). But the forms of w(x) and (j)(x) are now those given by equations (10) with j=3. If the warping constraint is included in the analysis the boundary conditions become eight. The six of those are again found by considering equations (10) with j=4 and substituting the resultant forms into equations (5) and (6). The remaining two can be found by evaluating equation (11) at both ends. When the flexural and torsional displacement expressions are substituted into the relevant equations, a set of equations is obtained. For the case of a load Pz and no warping constraint, the following equations can be found for j=3. EI^ w’”(0) + Kt,iw(0)=0 EI^ (E kn^ An+ (-1) PzZ - kn^ a „ 6 ' "f ' ) ^t,l ( E ■^n ■^PzE^n^ n f ) “0 (12) 1334 (13) EI^w”(0) -Kr,i w’(0)=0: EI^( I kn'An + Pz t kn'an 6 ‘ ' ) n«l n«t ■ ( S kn An + Pz^'kn^n® n f ) n*l n“l GJ^^l«=o-K,„,.i W)=0: dx GJ( S k„ T„ A„ + (-1) (Pz cy) t -k„ 'f „ a„ e ' ’‘f') fl«l 11'I -K,or,i ( i % A„ + (Pz Cy) i >?„ a„ e ')=0 diKx) I GJ-^Ix.l + K,„„ <KL)=0: dx GJ( 2k„>P„ A„ e“+ (PzCy) i-k„'P„a„ ) ns I 11=1 + K,„,.r (I % A„ + (Pz Cy) t Tn a„ e "f ')=0 El5W”(L)+ Kr,rW’(L)=0: El^CS k„^A„e^‘-+(Pz) i k„^a„ ) n-l ii’*) + Kz,r( i k„A„ e“+ (Pz) t-k„a„ e■^'‘-^')=0 n»l n=i (14) (15) (16) EI^ w’”(L)-Kt,rW(L)=0: Eiaz kn'Ane'n^ +(Pz) 2 - kn^ a „ e n '^ ) 11=1 Iia] -K,|(Z A„eV+(Pz) Za„e^''-V)=0 (17) lt=l 11=1 Here (-1) multipliers are included due to the symmetry and anti-symmetry effects. Those equations can be cast into the following matrix form. = -{Terms containing Pz} (18) 1335 An eighth order equation represents the necessary matrix equation for the determination of An values if the warping constraint is included in the analysis. In that case, equations (12), (13), (16) and (17) are valid with j=4. On the other hand equations (14) and (15) should be replaced by, (GJ^-Er„^^)L.o <l>(0)=0: dx dx (GJ( £ k„ % A„ + (-1) (P. Cy) £ -k„ a„ e * „' ’‘f') - nol n»l Er„( £ k„3 'f „ A„ + (- 1 ) (P^ Cy) £ -k„ = % a„ e * „ ' -f')) it«l IIS’! -K,„,,,(|;'P„A„+(P,Cy)i 'P„a„e-^'’‘f')=0 (19) cr A/T^=A. (GJ “ EFo j 3 )• x=L I^tor>r y(L) 0 . dx dx (GJ( Jkn'Fn A„ e’=„‘-+ (PjCy) i-kn'Pnan ' )- n=t Ii»l EFoC Xkn^ 'J'n A„ eV+ (PzCy) £-k„^'P„a„ ')) 11=11 n-l + Kior,, ( X % A„ + (Pz Cy) X % a„ e * „' \ ')=0 (20) n-] ii3| where j=4. The remaining two equations are found by considering the warping of the extreme ends. If the ends are free to warp the axial stress is zero, if the ends are not to warp the axial displacements are zero at both ends. No elastic constraints are imposed on end warping. If the left end is free to warp and the right end is not to warp, the required boundary conditions can be shown to be; u’(0)=0:(x kn^'Pn A„ + PzX 'i'.ian k„^e-^ ' ^') = 0 ( 21 ) u(L)=0 :(X k„ e^‘' % A„ + PzX (-k„) '‘-"f') = 0 (22) n=] list All the equations can be put into the following matrix form A.' A: A, • = -{Terms containing Pz} (23) A, A. A. A. 1336 Required An values are numerically found from equations (18) or (23). Their substitution to the appropriate forms of equations (10) and (11) yield the required responses at any point on the beam. 3.RESULTS AND DISCUSSION The theoretical model used in the study is shown in Figure 1 and has the following geometric and material properties: L=l(m), A =1.0*10"'(m\ h = 5.0*10-^(m), 15 = 4.17*10-* (m\ Cy = 15.625*10'* (m), J = 3.33*lO"'' (m*), =7.26*10'* (m"'), p= 2700 (kg/m*), r„ =2.85*10''* (m*), E = 7*10'“ (N/m*), G = 2.6* 10'“ (N/m*). Structural damping for torsional vibrations is included through, complex torsional rigidity as GJ*=GJ(l+z|3). For coupled vibrations, it is also included through the complex flexural rigidity as El^(H-z'n). First presented are the results for purely torsional vibrations. A bar assumed to have the given L, p, lo, G and J values is considered. The bar is then restrained at both ends by springs having the same torsional stiffness Ktor- A very low damping, p=10'^, is assigned and the resonance frequencies are precisely determined. It is found that, the introduction of a small Ktor introduces a very low valued resonance frequency. That fundamental frequency increases with increasing Ktor and as torsional constraint reaches to very high values, it approaches to the fundamental natural frequency of torsionally fixed-fixed beam. Table 1 gives the fundamental frequencies for a range of Ktor values and Figure 2 represents the fundamental mode shapes for selected Ktor values. Table 1: Uncoupled Fundamental Torsional Resonance Frequencies ((3=10’^,No Warping Constraint) Ktor [N1 Frequency [Hz] 0 0. 10'^ 1.606 10'' 5.035 io“ 14.683 10' 28.407 10^ 32.678 10^ 33.187 10^ 33.239 ;10*“ 33.245 1337 Figure 6 on the other hand represents the low frequency torsional receptances of the case in which the warping constraint is taken into consideration and the ends are free to warp. This graph is included in order to show the variation of fundamental torsional resonance frequencies for a range of Ktor values. _- K,3r= K„,= kK,„= 5 * 10 ': 1 * 10 ': 5 - 10 " 1 - 10 ' 5 - 10 ' 1*10® Figure 6. Fundamental Frequencies of Purely Torsional Vibrations ((3=0.01, x=0.13579[m],Warping Constraint Included, Ends are Free to Warp) The second part of the study investigates the characteristics of doubly-coupled vibrations. Now, the effects of each constraint are separately considered. A channel having the given parameters is supported at its ends by springs Kt, Kr, and Ktor- Warping constraint is included in the analysis and the ends are assumed to warp freely. First analyzed is the effects of Ktor- For this Kt=10^° [N/m] and Kr =10^° [N] are assigned at both ends of the channel and kept fixed throughout the study. Ktor is varied and the frequencies are shown in Table 2. Table 2. Effects of Ktnr in Doublv-counled Vibrations (Kt=10^° [N/m] and Kr=10^® [N], Warping Constraint Included) A: First Torsion Dominated Frequency [Hz] B: First Flexure Dominated Frequency [Hz] Ktor [Nl A B 10-^ 1.607 134.603 10'' 5.069 134.936 10° 15.620 138.258 10' 39.725 169.247 10^ 56.691 318.423 10^ 59.528 422.061 10^ 59.828 430.642 1338 0.0 0.2 0.4 0.6 0.8 1.0 ND LENGTH Figure 2. Fundamental Mode Shapes of Purely Torsional Vibrations ((3=0, No Warping Constraint) Then, the warping constraint To is included in the analysis and the results of purely torsional vibrations are presented again. The beam had the same Ktor values at both ends and the numerical values of the relevant parameters are taken to be those previously defined. Figure 3 represents the fundamental mode shapes for which the ends are free to warp, whereas Figure 4 shows the mode shapes of the case in which there is no warping at the ends. Figure 3. Fundamental Mode Shapes of Purely Torsional Vibrations (P=0, Warping Constraint Included, Ends are Free to Warp) (P=0, Warping Constraint Included, Ends Can Not Warp) Figure 5 is drawn to highlight the effects of end warping. Both ends of the channel are restrained with Ktor=l * lO’ [N] and all the other parameters of the study are kept fixed. Figure 5 represents the direct torsional receptances of two cases in which the ends of the channel are allowed to warp and not to warp in turn. It can be seen that the prevention of end warping increases the resonant frequencies. Figure 5. Frequency Response of Purely Torsional Vibrations ([3=0.01, X =0.13579 [m], Ktor=l*10' [N]» Warping Constraint Included) 1340 It can be seen that, when it has lower values Ktor is more effective on the torsion dominated resonance frequencies. For the higher Ktor values, the effects are more apparent on the flexure dominated frequencies. Figure 7 represents the direct flexural receptance of the channel for a set of selected end stiffnesses. Torsion dominated resonances at 59.528 [Hz], 206.071 [Hz] and 476.649 [Hz] appear as spikes. The flexure dominated resonance occurs at 422.061 Hz. Figure 7. Frequency Response of Doubly-coupled Vibrations (ti=0.001, P=0.001, x=0.13579[m], Warping Constraint Included, Ends are free to warp, Kr=1.10^°[N], Kt=1.10^° [N/m], Ktor=l-10^ [N]) Then the effects of the rotational spring, Kr, are considered. The ends of the channel are assumed to be restrained with Kt=10^° [N/m] and Ktor =10^° [N]. The resulting frequencies are given in Table 3 for a range of Kr values. Table 3. Effects of Kr in Doublv-coupled Vibrations (Kt=10^° [N/m] and Ktor=10^° [N], Warping Constraint Included) A; First Torsion Dominated Frequency [Hz] B: First Flexure Dominated Frequency [Hz]_ Kr[N] _A_ B 10*^ 58.678 205.344 10° 58.679 205.358 10^ 58.699 206.688 10^ 58.858 218.055 10" 59.437 289.738 10^ 59.799 405.667 10^ 59.855 429.345 1341 It is seen that Kr is not effective on torsion dominated resonance frequencies, but plays significant role for flexure dominated resonance frequencies. Finally considered the effects of the transverse spring Kt. Again, the channel is assumed to have very high Kr and Ktor values at both ends and Kt values are varied. Table 4 shows the resonance frequencies. Table 4. Effects of Kt in Doublv-coupled Vibrations (Kr=10^° [N] and Ktor=10^° [N], Warping Constraint Included) A: First Torsion Dominated Frequency [Hz] B: First Flexure Dominated Frequency [Hz]_ Kt [N/m] A B 10' 1.369 70.455 10^ 4.328 70.501 10^ 13.592 70.973 10^ 39.282 77.379 10= 58.095 159.665 10= 59.699 361.255 10* 59.860 431.215 It can be seen that the transverse stiffness, like torsional stiffness, effects both flexure and torsion dominated frequencies. 4.CONCLUSIONS In this study, a new analytical method is presented for the analysis of forced vibrations of open section channels in which the flexible supports provide the end constraints. The dynamic response of open section channels is a coupled problem and their analysis requires the simultaneous consideration of all the possible vibratory motions. The wave propagation approach is an efficient tool for this complicated problem and the developed method is based on that. The current method analyzes the forced, coupled vibrations of open section channels. The channels, taken as Euler-Bernoulli beams, have uniform cross- section and a single symmetry axis. That consecutively leads to the coupling of flexural vibrations in one direction and torsional vibrations. The excitation is assumed to be in the form of a harmonic point force, acting at the centroid. 1342 Various frequency response curves of uncoupled and coupled vibrations are presented for a variety of different elastic end boundary conditions( which may also have the inertial properties). The developed method, although aimed at determining the forced vibration characteristics, is also capable of determining the free vibration properties. This is also demonstrated by presenting various mode shape graphs. It has been determined that the transverse and the torsional stiffnesses play more significant role as compared to the rotational stifness. The method can be used in analyzing the effects of multi point and/or distributed loadings. This can simply be achieved by modifying the terms of the forcing vector without increasing the order of the relevant matrix equation. The developed method can also be used in the analysis of elastically supported, triply-coupled vibrations of uniform channels. Results of that study will be the subject of another paper. REFERENCES 1. Gere, J.M. and Lin, Y.K., Coupled Vibrations of Thin-Walled Beams of Open Cross-Section. J. AppliedMech Trans.ASME.,\9SZ, 80,373-8. 2. Lin, Y.K., Coupled Vibrations of Restrained Thin-Walled Beams. J. Applied Mech. Trans.ASME., 1960, 82, 739-40. 3. Dokumaci, E., An Exact Solution for Coupled Bending and Torsional Vibrations of Uniform Beams Having Single Cross-Sectional Symmetry. JSoundandVib.Am, 119,443-9. 4. Bishop, R.E.D, Cannon, S.M. and Miao, S., On Coupled Bending and Torsional Vibration of Uniform Beams. J.Sound and Fi'/).,1989,131,457-64. 5. Cremer, L. and Heckl, y\..,Structure~ Borne Sound, Springer-Verlag,1988. 6. Mead, D.J. and Yaman, Y., The Harmonic Response of Uniform Beams on Multiple Linear Supports: A Flexural Wave Analysis. J. Sound and Vib, 1990, 141,465-84 7. Yaman, Y. Wave Receptance Analysis of Vibrating Beams and Stiffened Plates. PA Z). Ttew, University of Southampton, 1989. 8 Yaman, Y., Vibrations of Open-Section Channels: A Coupled Flexural and Torsional Wave Analysis. (J. Sound and Vib, Accepted for publication) 1343 1344 THE RESPONSE OF TWO-DIMENSIONAL PERIODIC STRUCTURES TO HARMONIC AND IMPULSIVE POINT LOADING R.S. Langley Department of Aeronautics and Astronautics University of Southampton Southampton S017 IBJ ABSTRACT Much previous work has appeared on the response of a two-dimensional periodic structure to distributed loading, such as that arising from a harmonic pressure wave. In contrast the present work is concerned with the response of a periodic structure to localised forcing, and specifically the response of the system to both harmonic and impulsive point loading is considered by employing the method of stationary phase. It is shown that the response can display a complex spatial pattern which could potentially be exploited to reduce the level of vibration transmitted to sensitive equipment. 1. INTRODUCTION Many types of engineering structure are of a repetitive or periodic construction, in the sense that the basic design consists of a structural unit which is repeated in a regular pattern, at least over certain regions of the structure. An orthogonally stiffened plate or shell forms one example of an ideal two-dimensional periodic structure in which the fundamental structural unit is an edge stiffened panel. Although a completely periodic structure is unlikely to occur in practice, much can be ascertained regarding the structural dynamic properties of a real structure by considering the behaviour of a suitable periodic idealization. For this reason, much previous work has been performed on the dynamic behaviour of two-dimensional periodic structures, with particular emphasis on free vibration and the response to pressure wave excitation [1,2]. However, no results have yet been appeared regarding the response of two-dimensional periodic structures to point loading (as might arise from equipment mounts), and this topic forms the subject of the present work. A general method of computing the response to both harmonic and impulsive loading is presented, and this is then applied to an example system. Initially the response of a two-dimensional periodic structure to harmonic point loading is considered, and it is shown that the far-field response can be expressed very simply in terms of the "phase constant" surfaces which describe the propagation of plane waves. It is further shown that for 1345 excitation within a pass band two distinct forms of response can occur; in the first case the amplitude of the response has a fairly smooth spatial distribution, whereas in the second case a very uneven distribution is obtained and "shadow zones" of very low response are obtained. The second form of behaviour is related to the occurrence of caustics (defined in section 3.3), and the distinctive nature of the response suggests that a periodic structure might be designed to act as a spatial filter to isolate sensitive equipment from an excitation source. Attention is then turned to the impulse response of a two-dimensional periodic structure. It is again shown that the response can be expressed in terms of the phase constant surfaces which describe the propagation of plane waves. The application of the method of stationary phase to this problem has a number of interesting features, the most notable being the fact that four or more stationary points can arise. It is found that a surface plot of the maximum response amplitude against spatial position reveals features which resemble the "caustic" distributions obtained under harmonic loading. 2. RESPONSE TO A HARMONIC POINT LOAD 2.1 Modal Formulation and Extension to the Infinite System A two-dimensional periodic structure consists of a basic unit which is repeated in two directions to form a regular pattern, as shown schematically in Figure 1. Each unit shown in this figure might represent for example an edge stiffened curved panel in an aircraft fuselage structure, a three-dimensional beam assembly in a roof truss structure, or a pair of strings in the form of a "-f" in a cable net structure. The displacement w of the system can be written in the form w{n,x), where n={n^ nf) identifies a particular unit and x = {Xi X 2 X 3 ) identifies a particular point within the unit. The coordinate system x is taken to be local to each unit, and the precise dimension of both X and the response vector w will depend on the details of the system under consideration. The present section is concerned with the response of a two- dimensional periodic structure to harmonic point loading of frequency co. In the case of a system of finite dimension, the response at location (n,x) to a harmonic force F applied at (0 ,jCo) can be expressed in the standard form [3] »’(n,*)=EE P 9 ( 1 ) where rj is the loss factor, 4>p^(n,x) are the modes of vibration of the system and (j}pg are the associated natural frequencies. The modes ^p^ which appear in equation ( 1 ) are scaled to unit generalized mass, so that 1346 /!, «! V ( 2 ) where V represents the volume (or equivalent) of a unit and p(jc) is the mass density. The present concern is with the response of an infinite system, or equivalently the response of a large finite system in which the vibration decays to a negligible level before reaching the system boundaries. In this case the response is independent of the system boundary conditions, and it follows that any analytically convenient set of modes can be employed in equation (1). As explained in reference [4], it is expedient to consider the Born-Von Karman (or "periodic”) boundary conditions, as in this case the modes of vibration can be expressed very simply in terms of propagating plane wave components. In this regard it can be noted from periodic structure theory [5] that a propagating plane wave of frequency w has the general form w(rt,jc)=i?^{^(jc)exp(zej«, +ie^n^nu)t)}, (3) where and eo are known as the propagation constants of the wave (with - T<e,<T and - 7 r<G 2 <T for uniqueness), and g{x) is a complex amplitude function. By considering the dynamics of a single unit of the system and applying Bloch’s Theorem [5], it is possible to derive a dispersion equation which must be satisfied by the triad (w, 61 , 62 ) - by specifying Gj and €2 this equation can be solved to yield the admissible propagation frequencies w. By way of example, solutions yielded by this procedure for a plate which rests on a grillage of simple supports are shown in Figure 2 (after reference [ 6 ]). It is clear that the solutions form surfaces above the 61-62 plane - these surfaces are usually referred to as "phase constant" surfaces, and a single surface will be represented here by the equation a;= 0 ( 61 , 62 ). The phase constant surfaces always have cyclic symmetry of order two, so that 0 (ei, 62 )= 0 (- 6 i,- 62 ); for an orthotropic system the surfaces also have cyclic symmetry of order four, and therefore only the first quadrant of the 61-62 plane need be considered explicitly, as in Figure 2. The key point about the Born-Von Karman boundary conditions is that a single propagating wave can fully satisfy these conditions providing and 6 o are chosen appropriately. The conditions state that the left hand edge of the system is contiguous with the right hand edge, and similarly the top edge is contiguous with the bottom edge, so that the system behaves as if it were topologically equivalent to a torus. If the system is comprised of XN 2 units, then a propagating wave will satisfy these conditions if and 62 ^ 2 =2x^ for any integers p and q. Following equation (3), the displacement associated with such a wave can be written in the form 1347 ( 4 ) where ei^ and € 2 , are the appropriate values of the phase constants, and <^^^=0(ej^,e2g)- Now since it follows that a wave of frequency travelling in the opposite direction to will also satisfy the boundary conditions. This wave say) will have the form where it has been noted from periodic structure theory that reversing the direction of a wave leads to the conjugate of the complex amplitude function ^(x). The two waves represented by equations (4) and (5) can be combined with the appropriate phase to produce two modes of vibration of the system in the form KSn,x) • Re Im ( 6 ) By adopting this set of modes it can be shown [4] that equation (1) can be re¬ expressed as K.(«,x)= £ £ 2g;WF^g„(xJexp(-ie,^«,-%,n,) ^=1-^/2 9=1 -Nj/2 +Z?7) -( J ? where and M have been taken to be even, and the amplitude function gp^ is scaled so that g^,=[2p(x)WV,Ar,]-%(x), {llV)\^f„(x)f;,(x)dx=\, (8,9) where the normalized amplitude function fp^ is defined accordingly. The summation which appears in equation (7) includes only those modes associated with a single phase constant surface 0 (€i, 62 ); if more than one surface occurs then the equation should be summed over the complete set of surfaces. The summation will include modes for each surface, which is consistent with known results for the modal density of a two-dimensional periodic structure. Equation (7) yields the response of a finite system of dimension XM to a harmonic point load - this response is identical to that of an infinite system if the vibration decays to a negligible amount before meeting the system boundaries. If the system size is allowed to tend to infinity in equation 1348 (7) then neighbouring values of the phase constants e^p and € 2 ^ become closely spaced (since deip=ei_p+i-eip=2T/A^i and de 2 g=€ 2 , 9 + 2 x 77 / 2 )> and in this case the summations can be replaced by integrals over the phase constants to yield - LI me„e,)ninv)-c^^ where 0 )= 0 ( 61 , 62 ) and ^(x) is the complex amplitude associated with the wave (£ 0 , 61 , 62 ). The evaluation of the integrals which appear in equation (10) is discussed in the following sub-sections. 2.2 Integration over 61 The integral over 6 i which appears in equation (10) can be evaluated by using contour integration techniques. Two possible contours in the complex e, plane are shown in Figure 3; to ensure a zero contribution from the segment Im( 6 i) = ±oo, the upper contour is appropriate for /Zi <0 while the lower contour should be used for n^X). For each contour the contributions from the segments and 61 ;;=x cancel, since the integrand which appears in equation (10) is unchanged by an increment of 2x in the real part of ei. The only non-zero contribution to the integral around either contour therefore arises from the segment which lies along the real axis. The poles of the integrand occur at the 61 solutions of the equation [fl(6i,62)?(U/i7)-a;^=0, (11) for specified 62 and oj. By definition there will be two real solutions^ in the absence of damping ( 77 = 0 ) providing the frequency range covered by the phase constant surface includes oj. Any complex solutions to equation (11) in the absence of damping will correspond physically to "evanescent" waves which decay rapidly away from the applied load. The present analysis is concerned primarily with the response of the system in the far field (that is, at points remote from the excitation source), and for this reason attention is focused solely on those roots to equation (11) which are real when 77 = 0 . The effect of damping on these roots can readily be deduced: if 77 is small then it follows from equation ( 11 ) that a real solution 6 ^ will be modified to become 6 I-i(o 7 / 2 )( 5 Q/^ 6 l)■^ and hence the real pole for which dn/ 36 i <0 is moved to the upper half plane, while that for which 30/56, >0 is moved to the lower half plane. Given that the residue at such a pole is proportional to (30/36i)'\ ^One positive and one negative. These solutions will have the form ± 6 , for an orthotropic system. 1349 it follows that the sign of the residue which arises from the contour integral is determined by the integration path selected, and hence by the sign of These considerations lead to the result ~Tr~l fi|3fi/3«,|(l+/)/) ( 12 ) where e,(£ 2 ,M) is the appropriate solution to equation (11). The evaluation of the integral over €3 is discussed in the following section. 3.3 Integration over €3 Since the present concern is with the response of the system at some distance from the excitation point, the integral over e, which appears in equation ( 12 ) can be evaluated to an acceptable degree of accuracy by using the method of steepest descent [7], With this approach it is first necessary to identify the value of €2 for which the exponent -i(eirti+e 2 « 2 ) is stationary. The condition for this is {3ejde^n^+n^=0. (13) Now Gi and satisfy the dispersion relation, equation (11), and thus equation (13) can be re-expressed in the form (aQ/a62)«r(5^2/a€,)«2=o, (i4) where it has been noted from equation ( 11 ) that, for fixed co, 3 ei/ 3 € 2 =- ( 5 Q/ 3 e 2 )/( 9 fi/ 36 i), In the absence of damping the wave group velocity lies in the direction (SQ/Sei and in this case it follows from equation (14) that the group velocity associated with the required value of €2 is along («i ru). For light damping this result will be substantially unaltered, although damping will have an important effect on the value of the exponent -\{€^n^-\-e 2 iv^ at the stationary point. This effect can be investigated by noting initially that d{e^n^+&^n^)IBr}~{deJbri B&Jbr]).{n^ n^. (15) Now it follows from equation (11) that for light damping {ri<l) (dQ/de, dQlde,).{de,ldr] de^/dv) = -io)l2, (16) and hence equations (14)-(16) can be combined to yield the following result at the stationary point d{e^n^+e^n2)/dr) = -io)n/2c^. (17) 1350 Here Cg=^{{d£l!dex?+{d^lbe^'^] is the resultant group velocity and n=V[ni+ni-'\ is the radial distance (in units) from the excitation point to the unit under consideration. It follows that in the immediate vicinity of the stationary point the exponent can be expanded in the form (18) where the subscript 0 indicates that the term is to be evaluated at the stationary point under the condition 7 ]= 0 ; for ease of notation, this subscript is omitted in the following analysis. The method of steepest descent proceeds by substituting equation (18) into equation ( 12 ) and assuming that: (i) the main contribution to the integral arises from values of 62 in the immediate vicinity of the stationary point; (ii) the integrand is effectively constant in this vicinity, other than through variation of the term e 2 -(e 2 )o which appears in equation (18); (iii) under conditions (i) and (ii) the integration range can be extended to an infinite path without significantly altering the result. The method then yields [7] w{n,x) = ~if *F 7 o[ 20 V|aQ/a£j/ 2 xp(A:)p(A:o)|«i(aV 5 e 2 )| . (19) where/is the normalized complex wave amplitude which appears in equation (9), and all terms are to be evaluated at the stationary point. The stationary point associated with equation (19) is that point for which the group velocity is in the {n^ rQ direction. Geometrically, this is the point at which the normal to the curve a)=Q(ei, £ 2 ) itt the plane lies in the {n^ Ho) direction. Three such curves are shown schematically in Figure 4, together with a specified (n, 722 ) direction. For the frequencies and coj the situation is straight forward, in the sense that a unique stationary point exists for any {n^ n^) direction. For the frequency 0 J 2 the situation is more complex, since: (i) two stationary points occur for the (n^ rQ direction shown, and (ii) no stationary point exists if the {n^ direction lies beyond the heading B shown in the figure (the dashed arrow represents the normal with maximum inclination to the axis). In case (i) equation (19) should be summed over the two stationary points, while in case (ii) the method of steepest descent predicts that w{n,x) will be approximately zero, leading to a region of very low vibrational response. If the direction («i coincides with the dashed arrow, then equation (19) breaks down, since it can be shown that at this point. The heading indicated by the dashed arrow represents a caustic [7], and the theory given in the present section must be modified for headings 1351 (Wi Wj) which are in the immediate vicinity of the caustic - full details of the appropriate modifications are given in reference [4]. An example which illustrates the application of equation (19) is given in section 4. 3. RESPONSE TO AN IMPULSIVE POINT LOAD If the system is subjected to an impulsive (i.e. a delta function applied at r=0), rather than harmonic, point load, then equation (10) becomes [8] w{n,x,t)={N^NJ2Tp) f [ g* II (20) where co = ^](€i,€ 2 )- The method of stationary phase can be applied to this expression to yield [8] w{n,x,t)-{\l2yr)[p{x)p{x)\J\r‘^^^^ exp(-/Gjtti -k^n^nQt+ib )}, ( 21 ) where all terms are evaluated at the stationary point, and J and 5 are defined as /=(a"n/a6?)(a"Q/ae^-(3"0/ae,a6,)", S=(Tr/4)sgn(a%/fle?){l+sgn(/)}. (22,23) In this case the stationary point is given by the solution to the equations =(afi =(30 /3e,>. (24,25) In practice equations (24) and (25) may yield multiple solutions (stationary points), in which case equation (21) should be summed over all such points. Furthermore, stationary points having 7=0 indicate the occurrence of a caustic, and equation (21) must be modified in the immediate vicinity of such points as detailed in reference [8]. An example of the application of equation (21) is given in the following section. 1352 4. EXAMPLE APPLICATION 4.1 The System Considered The foregoing analysis is applied in this section to a two-dimensional periodic structure which consists of a rectangular grid of lumped masses m which are coupled through horizontal and vertical shear springs of stiffness and h respectively. Each mass has a single degree of freedom consisting of the out- of-plane displacement w, and a linear spring of stiffness k is attached between each mass and a fixed base. It is readily shown that the system has a single phase constant surface of the form Q -(ep€,)=iLii(l-cos€i)+/x2(l-cos€2)+a;^, where ix^^lkjm, and o)^-==klm. The function U can be used in conjunction with the analysis of the previous sections to yield the response of the system to harmonic and impulsive point loading; in this regard it can be noted that for the present case p{x)=m, V—l, and/(:»:) = 1. 4.2 Response to Harmonic Loading The surface is shown as a contour plot in Figure 5 for the case m=1.0, oj„==0, Ati = 1.0, Results for the forced harmonic response of this system at the two frequencies w = 1.003 and cx> = 1.181 are shown in Figures 6 and 7. In each case the response of a 40 x40 array of point masses is shown; a unit harmonic point load is applied to mass (21,21) and the loss factor is taken to be =0.05. Two sets of contours are shown in each Figure: the smooth contours have been calculated by using equation (19) while the more irregular contours have been obtained by a direct solution of the equations of motion of the finite 1600 degree-of-freedom system. By considering the results shown in Figure 6, it can be concluded that: (i) for the present level of damping the finite system effectively behaves like an infinite system, and (ii) the analytical result yielded by equation (19) provides a very good quantitative estimate of the far field response. It can be noted from Figure 5 that no caustic occurs for a) = 1.003, in the sense that equation (14) yields only one stationary point which contributes to equation (19). In contrast, a caustic does occur for the case a) = 1.181, and this leads to the very irregular spatial distribution of response shown in Figure 7. Two stationary points contribute to equation (19), and constructive and destructive interference between these contributions is responsible for the rapid fluctuations in the response amplitude. It is clear that the response exhibits a "dead zone" for points which lie beyond the caustic heading (in this case 30.25° to the «i-axis), as predicted by the analysis presented in section 2. 4.3 Response to Impulsive Loading The impulse response of a system having m=1.0, /xi = 1.0, ^2=0.51, and 1353 co„'“0.25 has been computed. The impulse was taken to act at the location /Zi=« 2=0 and the time history of the motion of each mass in the region - iO<(72i,n2) ^ 10 was found by using equations (21)-(23). For each mass the maximum response \w\ was recorded, and the results obtained are shown as a contour plot in Figure 8. In accordance with Fourier’s Theorem, the impulse response of the system contains contributions from all frequencies, and therefore the spatial distribution of | w| can be expected to lie somewhere between the two extreme forms of harmonic response exhibited in Figures 6 and 7. This is in fact the case, and the response shown in Figure 8 retains a distinctive spatial pattern. As discussed in reference [8], the results shown in Figure 8 are in good agreement with direct simulation of the impulse response of the system. 5. CONCLUSIONS This paper has considered the response of a two-dimensional periodic structure to both harmonic and impulsive point loading. With regard to harmonic loading, it has been shown that the spatial pattern of the response is strongly dependent on the occurrence of a caustic: if no caustic occurs then the response has a fairly smooth spatial distribution, whereas the presence of a caustic leads to an irregular spatial distribution and a "dead zone" of very low response. This type of feature is also exhibited, although to a lesser degree, in the spatial distribution of the response to an impulsive point load. This behaviour could possibly be exploited to reduce vibration transmission along a specified path, although the practicality of this approach for a complex system has yet to be investigated. The present analytical approach can be applied to all types of two-dimensional periodic structure - the information required consists of the phase constant surface(s) 0 ( 61 , 62 ) and the associated wave form(s) f{x), both of which are yielded by standard techniques for the analysis of free wave motion in periodic structures [1,2]. REFERENCES 1. S.S. MESTER and H. BENAROYA 1995 Shock and Vibration 2, 69- 95. Periodic and near-periodic structures. 2. D.J. MEAD 1996 Journal of Sound and Vibration 190, 495-524. Wave propagation in continuous periodic structures: research contributions from Southampton 1964-1995. 3. L. MEIROVITCH 1986 Elements of Vibration Analysis, Second Edition. New York: McGraw-Hill Book Company. 4. R.S. LANGLEY 1996 Journal of Sound and Vibration (to appear). The response of two-dimensional periodic structures to point harmonic forcing. 1354 1 ■ ■ ■ 1 ■ ■ ■ 0 m ■ 1 ■ Xj. ■j' Oi <- ^1 1 ^ Figure I, Schematic of a two-dimensional periodic structure. The arrow indicates' the reference unit (with n=0) while the circle represents a general point (re,x). The structure may have a third spatial coordinate X 3 , which for convenience is not shown in the figure. The point load considered in section 3 is applied at the location of the arrow. Figure 2. Phase constant surfaces for a plate which rests on a square grillage of simple supports. Q is a non-dimensional frequency which is def as Q=a)LV(m/D), where m and D are respectively the mass per unit area and the flexural rigidii^ the plate, and L is the support spacing. 1355 Figure 5. Contour plot of the phase constant surface for the case ^=0.57. The contours are separated by an increment Aco=0.1477. The two contours considered in section 4.2 are indicated as follows: (a) a) = 1.033; (b) a) = 1.181. Figure 6. Response |H'(n,x)|2 of the 40x4v .uass/spring system to a unit harmonic force of frequency w-1.033 applied at the location i=y=21. The contours correspond to the response levels lK«,x)!'=0.01, 0.02, andO.05. 1357 STICK-SLIP MOTION OF AN ELASTIC SLIDER SYSTEM ON A VIBRATING DISC HOuyang J E Mottershead M P CartmeU' MIFiiswell^ Department of Mechanical Engineering, University of Liverpool ^ Department of Mechanical Engineering, University of Edinburgh ^ Department of Mechanical Engineering, University of Wales Swansea ABSTRACT The in-plane vibration of a slider-mass which is driven around the surface of a flexible disc, and the transverse vibration of the disc, are investigated. The disc is taken to be an elastic annular plate and the slider has flexibility in the circumferential (in-plane) and transverse directions. The static fiiction coefi&cient is assumed to be higher than the kinetic friction. As a result of the fiiction force acting between the disc and the slider system, the slider will oscillate in the stick-slip mode in the plane of the disc. The transverse vibration induced by the slider will change the normal force of the slider system acting on the disc, which in turn will change the in-plane oscillation of the slider. For different values of system parameters, the coupled in-plane oscillation of the slider and transverse vibration of the disc will exhibit quasi-periodic as well as chaotic behaviour. Rich patterns of chaotic vibration of the slider system are presented in graphs to illustrate the special behaviour of this non-smooth nonlinear dynamical system The motivation of this work is to analyse and understand the instability and/or squeal of physical systems such as car brake discs where there are vibrations induced by non-smooth dry-fiiction forces. NOMENCLATURE a , b mner and outer radii of the annular disc c damping coefficient of the slider in in-plane direction h thickness of the disc i = V=T 1359 k, k transverse and in-plane stiffiiess of the slider system m r t ^stiek mass of the slider radial co-ordinate in cylindrical co-ordinate system radial position of the slider modal co-ordinate for k nodal circles and / nodal diameters time the time of the onset of sticking u , Mp transverse displacement of the slider mass and its initial value w , Wq transverse displacement of the disc and its initial value flexural rigidity of the disc Kelvin-type damping coefficient Young’s modulus initial normal load on the disc jfrom the slider system total normal force on the disc from the slider system combination of Bessel functions representing mode shape in radial direction circumferential co-ordinate of cylindrical co-ordinate system kinetic and static dry fiction coefficient between the shder and the D D' E N P R. 0 disc V P (P ^ stick ¥ ¥ki CO. Poisson’s ratio damping ratio of the disc specific density of the disc absolute circumferential position of the slider absolute circumferential position of the slider when it sticks to the disc circumferential position of the slider relative to the drive point mode function for the transverse vibration of the disc corresponding to ^kl natural (circular) frequency correq)onding to < 5 ^^ 1360 Q constant rotating speed of the drive point around the disc in radians per second INTRODUCTION There exists a whole class of mechanical systems which involve discs rotating relative to stationery parts, such as car brake discs, clutches, saws, computer discs and so on. In these systems, dry-friction induced vibration plays a crucial role in system performance. If the vibration becomes excessive, the system might fail, or cease to perform properly, or make offensive noises. In this paper, we investigate the vibration of an m-plane slider system, with a transverse mass-spring-damper, attached through an in-plane spring to a drive point which rotates at constant speed around an elastic disc, and the vibrations of the disc. Dry friction acts between the sHder system and the disc. Dry-friction induced vibration has been studied extensively [1-4]. For car brake vibration and squeal, see the review papers [5,6]. The stick-slip phenomenon of dry-friction induced vibration is studied in the context of chaotic vibration [7- 10]. Popp and Stelter [7] studied such motion of one and two degrees of freedom system and foimd chaos and bifiircation. They also conducted experiments on a beam and a circular plate (infinite number of degrees of freedom). These theoretical works are about systems of less than three degrees of freedom, and the carrier which activates the friction is assumed to be rigid. In this paper, we consider an elastic disc so that the transverse vibrations of the disc are important. As a result of including the transverse vibrations of the disc, rich patterns of chaos, which have not been reported previously are found. If there is only shding present at constant speed, the problem is reduced to a linear parametric analysis which was carried out for a pin-on-disc system in [11] and for a pad-on-disc system in [12,13]. m-PLANE OSCILLATION OF THE SLIDER SYSTEM As the drive point, which is connected to the shder-mass through an in-plane, elastic spring, is rotated at constant angular speed around the disc, the driven slider will undergo stick-slip oscillations. The whole system of the shder and the disc is shown in Figure 1. The equation of the in-plane motion of the slider system relative to the rotating drive point, in the sliding phase, is. 1361 ( 1 ) while in sticking, the equation of the motion becomes. The relationship between the relative motion of the slider system to the drive point and its absolute motion (relative to the stationary disc) is (p = Qt + y/, (3) We consider the foEowing initial conditions which are intended to simulate what happens in a disc brake. The slider system is at rest and there is no normal loading on the disc j&om the slider. Then a constant normal load is applied which causes transverse vibrations in the disc. At the same time, the drive point is given a constant angular velocity. Other initial conditions are possible, so that there is no loss of generality. First, sliding from the initial sticking phase occurs when, The slider will stick to the disc agaiu when, ju^P (during sliding), (5) or it will begin to sHde again iJ^ ij/ = , \k^rQtp-\ > (dxiring sticking). (6) Consequently, the slider system will stick and slide consecutively on the disc surface. TRANSVERSE VIBRATION OF THE ANNULAR DISC The equation of motion of the disc under the slider system is, ph^ + D'V^'w + DW = --5(r-r„)5i$-<p)P. (7) a/" r The total force P is the summation of initial normal load N and the resultant of the transverse motion w of the slider. Its expression is, P - N+ mu + cu-hkiu-u^). (8) Since it is assumed that the slider system is always in perfect contact with the disc, then, 1362 (9) u{t) = w{r„(p{t)A- Substitution of ecpiations (8) and (9) into (7) leads to. ph ^ — + /:)VV = --5(r - r, )6(^ - (p)[N + dt dt ^..dw d^w d^w .dw dw ( 10 ) k(w-wj\. Note that equation (10) is valid whether the slider system is sticking or sliding. When the slider sticks to the disc, equation (10) reduces to. + D-V‘ ^+£)W = --5(r -)5(5 -<p)x St St r (11) d'^W ^ M [A^+m—+ c—+ A:(w-w„)]. COUPLED VIBRATIONS OF THE SLIDER AND THE DISC Assume that the transverse motion of the disc can be represented by, M>{r,e,t)=ttii/,{r,0)q,it), ( 12 ) Jt=0 /=-<io and, where {r) is a combination of Bessel functions satisfying the boundary conditions in radial direction at the inner radius and outer radius of the disc. The modal functions satisfy the ortho-normality conditions ofj (14) 1363 Equations (10) and (11) can be simplified by being written in terms of the modal co-ordinates from equation (12). During sticking, the motion of the whole system of the slider and the disc can be represented by, N + 2^0)- —Z^„('-o)^«('-o)exp[i(5-0(»]x (15) ph.U ''=0 s=-«o {mq„+cq^+k{q^-q„{^)\). The sticking phase can be maintained i^ rA\¥\</^XJ^+- ^Z Z ^«(?')exp(i/?»)x {mq„+cq„+ k[q„ - (0)]} ]. While in sliding, the motion of the whole system can be represented by. %+2^a>^,qu+a>lq„ = ^^«('i)exp(-i/<»)- -rrri Z^„('-o)^«(n,)exp[i(5-0«!’]x pnO r=<i 5 ^=-oo H?,. +i2sw„ +isw„) + k{q„-qS'^)]}, r„{m{j/ + kw) = -/i,Sign((Z))[A'' +- ^Z Z KMx exp(LS(Z>)M?„ +i2s^„ +(ls^-5>^)9„] + c{q„ +>s(?g„)+*[!?„ -?„(o)]}]. The sliding phase can be maintained if, 1364 \K^M<mXN+-J^ t t K{r,)es.p{is(p)x ■yjphb ^=0^^ {iTiq„+i2sj>q„+(is^-s^j)^)qJ+ ( 20 ) c(?„ +isj^J + k[q„ -9„(0)]}], when, \j/ = ~0 or ^ = 0. (21) COMPUTING PROCESS As the shder system sticks and sHdes consecutively, the governing equations of the coupled motions of the whole system switch repeatedly from equations (15), (16) and (17) to equation (18), (19) and (20). TTie system is not smooth. Since the condition which controls the phases of the slider system itself depends on the motions, it is also a nonlinear system, whether is a constant or a function of relative speed ^. In order to get modal co-ordinates, we have to truncate the mfinite series in equation (12) to jfinite terms. Then numerical integration is used to solve equations (15), (18) and (19). Here a fourth order Rimge-Kutta method is used for second order simultaneous ordinary differential equations. Since equation (18) has time-dependent coefficients, time step length has to be very small. Constant time step lengths are chosen when the m-plane slider motion is well within the sticking phase or the shding phase in the numerical integration. As it is imperative that the time step should be chosen such that at the end of some time intervals the shder happens to be on the sticking-shding interfaces, we use a prediction criterion to choose next time step length when approaching these interfaces. Therefore, at the sticking-shding interfaces, the time step length is variable (actuahy smaher than it is while weh within sticking or shding). Nevertheless, tbe interfaces equations ( equation (17) or equations (20-21)) are only approximately satisfied [10]. When transverse motion of the shder system becomes so violent that the total normal force P in it becomes negative or becomes several times larger than the initial normal load 7/, we describe the system as being unstable. Then the motion begins to diverge. But this instabihty should be distinguished from a chaotic motion which is bounded but never converge to a point. 1365 NUMERICAL EXAMPLES The following data are used in the computation of numerical examples: a = 0.065m, b = 012m, OTm,/z = 0.001m; = 120GPa, v= 0.35, Z)*= 0.00004; yW, =0.4, //^=0.24, A: = lOOON/m, = lOON/m, m = 0.1kg, p = 7000kg/m^ The disc is clamped at inner radius and free at outer radius. Note that in these numerical examples, the disc thickness is dehherately taken to be very small in order to reduce the amount of computing work. However, this will not affect the qualitative features of the results or conclusion drawn from the results thus obtained. The first five natural (circular) frequencies are 451.29, 462.73, 426.73, 508.23, 508.23. We will concentrate on the vibration solutions at different levels of initial normal load. But occasionally solutions at different rotating speed or different damping ratios are investigated. Unless specified expressly, the Poincare sections are for the in-plane vibration of the slider system First of all, we study the effect of the normal loadA^. Take f2=l0 and ^— When N is very small^ the Poincare section is a perfect ellipse which indicate the in-plane vibration of the slider system is quasi-periodic, as the transverse vibration of the disc is too small to affect total normal force P . A typical plot of such motion is shown in Figure 2 for A^=0.5kPa. As N increases, the sticking period gets longer, the bottom part of the elhpse evolves into a straight line, indicating phase points within the sticking phase. One of such plots is given in Figure 3 for N =3kPa. A further increase of N not only lengthens the straight line part of the Poincare section, but also creates an increasingly ragged outline in the arch part of the plot. The curve is no longer smooth and it seems that the in-plane motion begins to enter a chaotic state from the quasi-periodic state. Figure 4 presents the Poincare section plot for N =7.5kPa. There is a transition period from quasi-periodic motion to chaotic motion, extending from N =6kPa up to N =9kPa. Chaos becomes detectable at iV=10kPa, whose Poincare section is shown in Figure 5. Then chaotic vibration follows. When N ~15kPa, the arch part of the Poincare section becomes so fuzzy and thick that it should no longer be considered as a curve, but rather a narrow (fractal) area. A hlow-up’ view of the arch part reveals that phase points are distributed across the arch. Both plots are ^own in Figure 6. Between iV'=17.ikPa and 18.325kPa, the vibration of the slider enters a new stage, with Poincare sections looking like star clusters as illustrated in Figures 7 and 8. This kind of motions are rather extraordinary and 1366 have not been reported in other works on stick-shp motions with a rigid carrier. Afterwards, the ‘arch-door’ hke Poincare sections come back (see Figure 9). The difference from previous Poincare sections of lower N is that the new Poincare sections look like overlapping of earlier Poincare sections, which indicates a clear layered structure, as diown in Figure 10 and more obviously in the left hand side of Figure 11. At this stage, the vibration is very chaotic. To give the reader a better picture, the Poincare section of a fixed point on the disc at (= 0.1m and = 0), is also shown in the right hand side of Figure 11. The Poincare sections of the slider-mass and a point on the disc are also given in Figures 12-15. In Figure 12 for A^=30.5kPa, the vibration goes unstable. Here again, the Poincare sections have not been reported elsewhere. If disc damping is increased, vibration will become more regular, as shown in Figure 13. Comparing Figures 11 with 13, we see that increase of disc damping makes the vibration more concentrated though not always smaller. Unstable vibration can be stabilised with more disc damping, as seen from Figure 14. If there is no damping at aU, the resulting vibration due to dry fiiction will be unstable, even at very small normal load N . In Figure 15, the motion of the slider tends to run away in the tangential direction from the normal ellipse attractor, while the motion of the disc goes unbounded. Increasing the speed of the drive point seems to make vibration more chaotic and more unstable, as shown in Figures 16-18. At this stage, however, we are unable to make a definite conclusion on rotating speed as there might be intervals of regular motions and intervals of chaotic motion for . More numerical examples must be computed to draw a positive conclusion on this parameter. The correlation dimension is not a good measure of the vibration for the current problem because its values fluctuate in some numerical examples. This failure was perhaps first discovered in [7]. The reason can be either that the system is non-smooth, or that the system has multiple degrees of freedom, or both. Therefore, the correlation dimension or any other fractal dimensions is not presented in this paper. CONCLUSIONS In this paper, we studied the in-plane stick-slip vibration of a slider system with a transverse mass-spring-damper driven around an elastic disc through a spring from a constant speed drive point, and transverse vibrations of the disc. The 1367 whole system had been reduced to six degrees of freedom after simplification. From numerical examples computed so far, we can conclude that: 1. Both vibrations are very complex as this is a multi-degree of freedom, non¬ smooth system Rich patterns of chaotic vibration are found. Some have not been reported elsewhere. 2. For the normal pressure parameter, smaller values allow quasi-peiiodic solutions. Greater pressures result in chaotic motions. At certain large pressures, the vibrations become unstable. 3. Disc dartping makes vibration more concentrated to smaller areas and when sufficiently large it can stabilise otherwise unstable vibration. 4. An increase in the rotating speed can make the vibration more chaotic or more unstable. 5. Correlation dimension is not a good measure of the vibration of this multi¬ degree of freedom, non-smooth dynamical system 6. Much more investigation needs to be earned out in understanding and characterising the vibration of multi-degree of freedom, non-smooth dynamical systems. ACKNOWLEDGEMENT This research is supported by the Engineering and Physical Sciences Research Council (grant niunber J35177) and BBA Friction Ltd. REFERENCES 1. Nakai, M., Chiba, Y. and Yokoi, M., Railway wheel squeal. Bulletin of JSME, 1984, 27, 301-8 2. Lin, Y-Q and Wang Y-H, Stick-sHp vibration of drill strings. XEng.Ind., TransASME, 1991,113, 38-43 3. Ferri, A. A. and Bindemann, A.C., Damping and vibration of beams with various types of fiictional support conditions. J. VibAcoust, TransASME^ 1992, 114, 289-96 4. Lee, A.C., Study of disc brake noise using multi-body mechanism with friction interface. In Friction-Induced Vibration, Chatter, Squeal, and Chaos, Ed. Ibrahim, KA. and Soom, A., DE-Vol. 49, ASME 1992, pp.99- 105 1368 5. Ibrahim, R.A., Friction-induced vibration, chatter, squeal, and chaos. In Friction-Induced Vibration, Chatter, Squeal, and Chaos, Ed. Ibrahim, KA. and Soom, A, DE-VoL 49, ASME 1992, pp. 107-38 6. Yang, S. and Gibson, R.F., Brake vibration and noise: reviews ,comments, and proposed considerations. Proceedings of the 14th Modal Analysis Conference, The Society of Experimental Mechanics, Inc., 1996, pp.l342- 9 7. Popp, K and Stelter, P., Stick-slip vibration and chaos. Phil. Trans. R Soc. Lond. A(1990), 332, 89-106 8. Pfeiffer, F. and Majek, M., Stick-slip motion of turbine blade dampers. Phil. Trans. R Soc. Lond. A(1992), 338, 503-18 9. Wojewoda, J., Kapitaniak, T., Barron, R. and Brindley, J., Complex behaviour of a quasiperiodically forced experimental system with dry friction. Chaos, Solitons and Fractals, 1993, 3, 35-46 10. Wiercigroch, M., A note on the switch function for the stick-slip phenomenon. J.SoundVib., 1994,175, 700-4 11. Chan, S.N., Mottershead, J.E. and Cartmell, M.P., Parametric resonances at subcritical speeds in discs with rotating frictional loads. Proc. Instn. Mech. Engrs, 1994, 208, 417-25 12. Mottershead, J.E., Ouyang, R, Cartmell, M.P. and Friswell, M.I., Parametric Resonances in an annular disc, with a rotating system of distributed mass and elasticity; and the effects of friction and dan:q)ing. Proc. Royal Soc. Lond. A., 1997, 453, 1-19 13. Ouyang, H., Mottershead, J.E., Friswell, M.l. and Cartmell, M.P., On the prediction of squeal in automotive brakes. Proceedings of the 14th Modal Analysis Conference, The Society of Experimental Mechanics, Inc., 1996, pp. 1009-16 Figure 1. Slider system and disc in cylindrical co-ordinate system 1369 Figure 9. iV=21kPa Figure 10. iV=24kPa 1374 A Finite Element Time Domain Multi-Mode Method For Large Amplitude Free Vibration of Composite Plates Raymond Y. Y. Lee, Yucheng Shi and Chuh Mei Department of Aerospace Engineering Old Dominion University, Norfolk, VA 23529-0247 Abstract This paper presents a time-domain modal formulation using the finite element method for large-amplitude firee vibrations of generally laminated thin composite rectangular plates. Accurate fi'equency ratios for fundamental as well as higher modes of composite plates at various maximum deflections can be determined. The selection of the proper initial conditions for periodic plate motions is presented. Isotropic beam and plate can be treated as special cases of the composite plate. Percentage of participation from each linear mode to the total plate deflection can be obtained, and thus an accurate frequency ratio using a minimum number of linear modes can be assured. Another advantage of the present finite element method is that the procedure for obtaining the modal equations of the general DujB5ng-type is simple when compared with the classical continuum Galerkin’s approach. Accurate frequency ratios for isotropic beams and plates, and composite plates at various amplitudes are presented. Introduction Large amplitude vibrations of beams and plates have interested many investigators [1] ever since the first approximate solutions for simply supported beams by Woinowsky-Kiieger [2] and for rectangular plates by Chu and Herrmann [3] were presented. Singh et al. [4] gave an excellent review of various formulation and assumptions , including the finite element method for large amplitude firee vibration of beams. Srirangaraja [5] recently presented two alternative solutions, based on the method of multiple scales (MMS) and the ultraspherical polynomial approximation (UPA) method, for the large amplitude firee vibration of a simply supported beam. The fi'equency ratios for the fundamental mode, ca/C0L> at the ratio of maximum beam deflection to radius of gyration of 5.0 (Wmax/r =5.0) are 3.3438 and 3.0914, using the MMS and the UPA method, respectively. Eleven firequency ratios including nine firom reference [4] were also given (see Table 1 of reference [5]). It is rather surprising that the firequency ratio for the fundamental mode at Wn«x/r =5.0 for a simply supported beam varied in a such wide range: fi-om the lowest of 2.0310 to the highest of 3.3438, and with the elliptic function solution by Woinowsky-Kiieger [2] giving 2.3501. Similar wide spread exists for the vibration of plates. Rao et al [6] presented a finite element method for the large amplitude firee flexural vibration of unstiffened plates. For the simply supported square plate. 1375 frequency ratios from six different approaches were reported (see Table 1 of reference [6]). The frequency ratio at Wmax/h =1.0 varied from a low of 1.2967 to a high of 1.5314, with Chu and Herrmann’s analytical solution [3] at 1.4023. This paper presents a finite element time domain modal formulation for the large amplitude free vibration of composite plates. The formulation is an extension from the isotropic plates [7], and the determination of initial conditions for periodic motions was not employed in reference [7]. The convergence of the fundamental frequency ratio is investigated for a simply supported beam and a simply supported square plate with a varying number of finite elements and a varying number of linear modes. Accurate frequency ratios for fundamental and higher modes at various maximum deflections, and percentages of participation from various linear modes, are obtained for beams and composite plates. Formulation Strain-Displacement and Constitutive Relations The von Karman strain-displacement relations are applied. The strains at any point z through the thickness are the sum of membrane and change of curvature strain components: {e} = -w.xx V,y >+< / 2 > + z< -W,yy U,y+V,x^ W,yj = {Em} + {sb} + z{K} ( 1 ) where and {eb}are the membrane strain components due to in-plane displacements u and v and the transverse deflection w, respectively. The stress resultants, membrane force {N} and bending moment {M}, are related to the strain components as follows: ■[A] IB]' [B1 [Di; ( 2 ) where [A] is the elastic extensional matrix, [D] is the flexural rigidity matrix, and [B] is the extension coupling matrix of the laminated plate. Element Displacements, Matrices and Equations Proceeding from this point, the displacements in equation (1) are approximated over a typical plate element , e.g. rectangular [8] or triangular [9], using the corresponding interpolation functions. The in-plane displacements and the linear strains are interpolated from nodal values by 1376 where [Hm] and [Bm] denote the displacement and strain interpolation matrices, respectively, and {Wm} is the in-plane nodal displacement vector. The transverse displacement, slopes and curvatures are interpolated from the nodal values by w = [Hb]{w^}, |^’^| = [G]{wb}, {K} = [B|,]{wb} (4a,b,c) where [HJ and [G] and [BJ denote the bending displacement, slope and curvature interpolation matrices, respectively, and {wb } denotes the nodal transverse displacements and its derivatives. Through the use of Hamilton’s principle, the equations of motion for a plate element undergoing large amplitude vibration may be written in the form [hb] 0 T [H] [kB]l r[klNm]+[klNB] [klbml ,r[k2b] < [k»]J L ^'"*1 0 J ' [ 0 ( or [m]{ w} + m + [kl] + [k2] }{w} = 0 (5) where [m] and [k] are constant matrices representing the element mass and linear stiffiaess characteristics, respectively; Pel] and Pe2] are the first order and second order non-linear sfiffiiess matrices, respectively; PcInJ depends linearly on unknown membrane displacement ({Nm }= [A][Bni]{wni}); PcInb] depends linearly on the unknown transverse displacement ({NB}=[B][Bb]{Wb}); DelbnJ depends linearly on the unknown plate slopes and represents coupling between membrane and bending displacements; and [k2b] depends quadratically on the unknown plate slopes. System Equations After assembling the individual finite elements for the complete plate and applying the kinematic boundary conditions, the finite element system equations of motion for the large-amplitude free vibration of a thin laminated composite plate can be expressed as [M]{w}4- ([K]+ [K1(W)] + [K2(W)]){W} = 0 (6) where [M] and [K] are constant matrices and represent the system mass and stifiBness respectively; and [Kl] and [K2] are the first and second order nonlinear stifi&iess matrices and depend linearly and quadratically on the unknown structural 1377 nodal displacements {W}, respectively. Most of the finite element large amplitude fi-ee vibration results for plates and beams in the literature, e.g. references [1,6] and others, were based on eq. (6) using an iterative scheme and various approximate procedures. The system equations are not suitable for direct numerical integration because: a) the nonlinear stif&iess matrices [Kl] and [K2] are functions of the unknown nodal displacements, and b) the number of degrees of fireedom (DOF) of the system nodal displacements {W} is usually too large. Therefore, eq. (6) has to be transformed into modal or generalized coordinates followed by a reduction of the number of DOF. In addition, the general DufiSng-type modal equations will have constant nonlinear modal stifihiess matrices. This is accomplished by a modal transformation and truncation r=l where and cou are the natural mode (normalized with the maximum component to unity) and linear firequency from the eigen-solution ©|_j.[M]{ (|))«=[K]{ (|)}®. The nonlinear stififtiess matrices [Kl] and [K2] in eq. (6) can now be expressed as the sum of the products of modal coordinates and nonlinear modal stiffaess matrices as [Kl] = ^q,(t)[Kl((])«)] (8) r=I n n [K2] = ^ ^ qr (t)qs (t)[K2((|)(''), )] (9) r=l s=l The nonlinear modal stifihiess matrices [Kl]^"^ and [O]^'^^ are assembled from the element nonlinear modal stif&iess terms [kl]^^^ and Pc2]^“^ as ([K1]«.[K2]‘“))= ^([klf>,[k2]'“>) (10) al! elements + bdy. conds. where the element nonlinear modal stif&iess matrices are evaluated with the known linear mode Thus, the nonlinear modal stiffiiess [Kl]^^^ and [K2]^'®^ are constant matrices. Equation (6) is thus transformed to the general Duf6ng-type modal equations as 1378 ( 11 ) [M]{q}+([K]+[KlJ+[K2„]){q} = 0 where the modal mass and linear stifhiess matrices are diagonal ([M],[K]) = [<I.f([M],[K])[<I.] and the quadratic and cubic terms are LI [Kl,]{q} = [<Df J^qjKl] 1 ^) Vt=1 n n (rs) r=:l s=l \Wk} ( 12 ) (13) (14) AH modal matrices in eq. (11) are constant matrices. With given initial conditions, the modal coordinate responses {q} can be determined from eq. (11) using any direct numerical integration scheme such as the Runge-Kutta or Newmark-P method. Therefore, no updating of the vibration modes is needed [10]. For periodic plate osciHations have the same period T, the response of all modal coordinates should also have the same period T. Since the initial conditions wHl affect greatly the modal response, the determination of initial conditions for periodic plate osciHations is to relate each of the rest modal coordinates in powers of the dominated coordinate as arqi(t;IC) + brqi(t;IC) + Crq^(t;IC)+.= qr(t;IC), r = 2,3. ...n (15) where the 2k, br, Cr , . are constants to be determined, and IC denotes initial conditions. For a three-mode (n=3) system, it is accurate enough to keep up to the cubic term only in eq. (15) and this leads to two set of equations a2qi(tp;A,B,C) + b2qf(tp;A,B,C) + C2qJ(tp;A,B,C) = q2(tp;A,B,C), p = 1,2,3 a3qi (tp;A, B,C) + bsqJ (tp; A, B. C) + CsqJ (tp;A. B, C) = q3 (tp;A, B, C), p = 1,2,3 (16a,b) in which the modal coordinates qi, q 2 and qs at tp are known quantities and the initial conditions are qi(0)=A, q2(0) =B, q3(0)=C and qi(0) = q2(0) = q3(0) = 0. PracticaHy, only eight equations are needed to determine the eight unknowns a 2 , as. 1379 hi, b 3 , C 2 , C 3 , B and C through an iterative scheme. However, the number of equations can be more than the number of unknowns for accurate determination of initial conditions and the least square method is employed in this case. The time history of the plate maximum deflection can be obtained from eq. (7). The participation value from the r th linear mode to the total deflection is defined as maxjqj. n ^max|qi i=l (17) Thus, the minimum number of the linear modes for an accurate and converged frequency solution can be determined based on the modal participation values. Results and Discussions Assessment of Single-Mode Elliptic Function Solution The fundamental frequency ratio Co/col = 2.3051 at Wmax/r =5.0 for a simply supported beam obtained by Woinowsky-Kiieger [2] using a single-mode and elliptic fimetion is assessed first. The conventional beam element having six (four bending and two axial) DOF is used. A half-beam is modeled with 10, 15, 20 elements, and the lowest four symmetrical linear modes are used in the Duffing modal equations. Table 1 shows that a 20-element and 1-mode model will yield a converged result. The percentages of participation from each mode for various values of Wmax/r are given in Table 2. The modal participation values demonstrate that a single mode (n=l) will yield an accurate fundamental frequency because the contribution from higher linear modes to the total deflection is negligible (< 0.01 % for Wmax/r up to 5.0). There is a small difference in frequency ratios between the present finite element and the elliptic integral solutions. This is due to the difference between the axial forces of the two approaches, the finite element method (FEM) gives a non-uniform axial force in each element; however, the average value of the axial force for each element is the same as the one in the classic continuum approach. The lowest (2.0310) and the highest (3.3438) frequency ratios at Wmax/r =5.0 in reference [5] are not accurate. Frequency ratios for higher modes of the simply supported beam are obtained next. A model with 40-elements and 3-anti-symmetric modes for the whole beam is employed for the frequency ratio of the second nonlinear mode. The mode participations shown in Table 2 indicate that a single-mode approach will yield accurate frequency results. And the frequency ratios for the second mode are the same as those of the fundamental one. Thus, the present method agrees extremely well with Woinowsky-Krieger’s classic single-mode approach. 1380 The time history of the first two symmetric modal coordinates and the beam central displacement, phase plot, and power spectral density (PSD) at maximum beam deflection W^Jr = 5.0 for the fimdamental firequency (or mode) are shown in Fig. 1. The time scale is non-dimensional and Ti is the period of the fundamental linear resonance. It is noted that although the central displacement response looks like a simple harmonic motion, it does have a small deviation fi:om pure harmonic motion due to the second small peak in the spectrum. This is in agreement with classical solution that the ratio of the jfrequency of the second small peak to that of the first dominant peak is 3. Now we are ready to assess the single-mode fundamental firequency of a simply supported square plate obtained by Chu and Herrmann [3]. A quarter of the plate is modeled with 6 x 6, 7 x 7, 8 x 8 and 9 x 9 mesh sizes and 1, 2, 4 or 5 symmetrical modes are used. The conforming rectangular plate element with 24 (16 bending and 8 membrane) DOF is used. The in-plane boundary conditions are u = v = 0 on all four edges. Table 3 shows that the 8 x 8 mesh size in a quarter-plate and 4- mode model should be used for a converged and accurate frequency solution. Table 4 shows the fi:equency ratios and modal participation values for the lowest three modes at various Wmax/h for a simply supported square plate (8x8 mesh size in a quarter-plate). It indicated that at least two linear modes are needed for an accurate frequency prediction at Wmax/h =1.0, and the contribution of higher linear modes increase with the increase of plate deflections. The modal participation values also show that the combined modes (1,3)-(3,1) and (2,4)-(4,2) are independent of the large-amplitude vibrations dominated by (1,1) and (2,2) modes, respectively. The time history, phase plot, and PSD at the maximum deflection Wmax/h =1.0 for the fundamental mode are shown in Fig. 2a, and Tu is the period of the fundamental linear resonance. There is one small peak in the spectrum and the frequency ratio of the second small peak to the first dominant one is 3. The low (1.2967) and the high (1.5314) frequency ratios at Wmax/h =1.0 given in reference [6] are not accurate. The influence of the initial conditions on periodic motion is demonstrated in Fig. 2a and 2b. In Fig. 2a, the modal coordinates all have the same period, and the initial conditions are determined fi*om eq.(15). They are qii(0)/h=1.0, qi3+3i(0)/h= - 0.0155, qi 3 . 3 i (0)/h = 0.0, q33(0)/h=0.000813, and qi 5 + 5 i( 0 )/h= 0.00011, and initial velocities are null, whereas in Fig. 2b, qii(0)/h=1.0 and all others are nuU. The modal coordinates do not have the same period. Clamped Beam It is thus curious to find out whether multiple-mode is required for the clamped beam. Convergence study of the fundamental firequency ratios at Wmax/r =3.0 and 5.0 shown in Table 5 indicates that a 25-element (half-beam) and 4-mode model win yield accurate and converged results. The time history, phase plot and PSD at Wmax/r =5.0 are shown in Fig. 3. The modal participation values in Table 6 and the PSD in Fig. 3 confirm that at least two modes are needed for accurate firequency results. 1381 Symmetric Composite Plate A simply supported eight-layer symmetrically laminated (0/45/-45/90)s composite plate with an aspect ratio of 2 is investigated. The graphite/epoxy material properties are as follows; Young’s moduli Ei = 155 GPa, E 2 = 8.07 GPa, shear modulus Gn = 4.55GPa, Poisson’s ratio V 12 =0.22, and mass density p = 1550 kg/m^ A 12 X 12 mesh is used to model the plate. The in-plane boundary conditions are fixed (u=v=0) at all four edges. The first seven linear modes are used as the modal coordinates. Table 7 gives the fundamental firequency ratios and mode participation values for the linear modes in increasing firequency order. The modal participation values indicate clearly that four modes are needed in predicting the nonlinear fi-equency, and other three of the seven are independent of the fundamental nonlinear mode. Figure 4 shows the time-history, phase plot, and PSD at Wmax/h =1.0. UNS YMMETRIC COMPOSITE PLATE A simply supported two-layer laminated (0/90) composite plate of 15 x 12 x 0.048 in. (38 x 30 x 0.12 cm) is investigated. The graphite/epoxy material properties are the same as those of the symmetric composite plate. A12 x 12 mesh is used to model the plate. The in-plane boundary conditions are fixed at all four edges. The first four linear modes are used as the modal coordinates. Table 8 gives the fundamental firequency ratios and mode participation values for the linear modes in increasing fi-equency order. From the phase plot, the time histories and PSD shown in Fig. 5, it can be seen that the total displacement response has a non-zero mean (i.e. the positive and negative displacement amplitudes for all modal coordinates are not equal). The quasi-ellipse in the phase plot is not symmetrical about the vertical velocity-axis. In the PSD at Wmax/h =1.0, it is observed that there are four small peaks in the spectrum and the firequency ratios of the second, third, fourth and fifth peak to the first dominant one are 2, 3, 4 and 5, respectively. This observation indicates that the displacement response includes the superharmonances of orders 2, 3, 4, and 5. The curves, which the positive and negative displacement amplitudes are plotted against the fundamental firequency ratio, are also given in Fig. 5. The difference between the positive and negative amplitudes increases as the firequency ratio increasing. Conclusions A multimode time-domain formulation, based on the finite element method, is presented for nonlinear firee vibration of composite plates. The use of FEM enables the present formulation to deal with composite plates of complex geometries and boundary conditions, and the use of the modal coordinate transformation enables to reduce the number of ordinary nonlinear differential modal equations to a much smaller one. The present procedure is able to obtain the general Duffing-type modal equations easily. Initial conditions for all modal coordinates having the same time 1382 period are presented. The participation value of the linear mode to the nonlinear deflection is quantified ; they can clearly determine the minimum number of linear modes needed for accurate nonlinear frequency results. The present fundamental nonlinear fi-equency ratios have been compared with the single-mode solution obtained by Woinowsky-Kneger for simply supported beams and by Chu and Herrmann for simply supported square plates. The Woinowsky-Krieger’s single-mode solution is accurate. For all other solutions, however, two or more modes are needed. The nonlinear firequencies for symmetrically and unsymmetrically laminated rectangular composite plates are also obtained. The phase plot and power spectral density showed that nonlinear displacement responses are no longer harmonic, and multiple modes are required for isotropic clamped beams and isotropic and composite plates. References 1. M. Sathyamoorthy 1987 Applied Mechanics Review 40, 1553-1561. Nonlinear vibration analysis of plates: A review and survey of current developments. 2. S. Woinowsky-Kreger 1950 Journal of Applied Mechanics 17, 35-36. The effect of an axial force on the vibration of hinged bars. 3. H. N. Chu and G. Herrmann 1956 Journal of Applied Mechanics 23, 523-540. Influence of large amplitudes on jfree flexural vibrations of rectangular elastic plates. 4. G. Singh, A. k. Sharma and G. V. Rao 1990 Journal of Sound and Vibration 142, 77- 85. Large amplitude free vibration of beams-discussion of various formulations and assumptions. 5. H. R. Srirangaraja 1994 Journal of Sound and Vibration 175, 425-427. Nonlinear free vibrations of uniform beams. 6. S. R. Rao, A. H. Sheikh and M. Mukhopadhyay 1993 Journal of the Acoustical Society of America 93 (6), 3250-3257. Large-amplitude finite element flexural vibration of plates/stiffened plates. 7. Y. Shi and C. Mei 1996 Journal Sound and Vibration 193, 453-464. A finite element time domain modal formulation for large amplitude free vibration of beams and plates. 8. K. Bogna:, R. L. Fox and L. A. Schmit 1966 Proceeding of Conference on Matrix Methods in Structural Mechanics, AFFDL-TR-66-80, Wright-Patterson Air Force Base, Ohio, October 1965, 397-444. The gena-ation of interelement compatible stiffness and mass matrix using the interpolation formulas. 9. Teseller and T. J. R. Hughes 1985 Computer Methods in Applied Mechanics and Engineering 50, 71 -101. A three node Mindlin plate element with improved fransverse shear. 10. A. K. Noor 1981 Composites and Structures, 13, 31-44. Recent advances in reduction methods for nonlinear problems. 1383 Table 1. Convergence of the fundamental frequency ratio at Wmax/r =5.0 for a simply supported beam No. of elements and 4 modes (C0/(0l)i No. of modes and 20 elements (G)/G)l)i 10 2.3537 1 2.3506 15 2.3511 2 2.3506 20 2.3506 3 2.3506 -- 4 2.3506 Table 2 The lowest two frequency ratios and the modal participations for a simply supported beam W^nax/r Elliptic integral [2] (CO/OOl)! FEM Modal Participation % (CO/COl)! __ 0.2 1.0038 1.0038 100.00 0.000 0.000 0.4 1.0150 1.0149 100.00 0.000 0.000 0.6 1.0331 1.0331 100.00 0.000 0.000 0.8 1.0580 1.0581 100.00 0.000 0.000 1 1.0892 1.0892 100.00 0.000 0.000 2 1.3178 1.3179 100.00 0.002 0.000 3 1.6257 1.6258 100.00 0.004 0.000 4 1.9760 1.9761 99.99 0.005 0.000 5 2.3501 2.3506 99.99 0.009 0.000 W.„ax/r (co/g)l)2 (C0/CDl)2 q.2_ _ q4 Q6 0.2 1.0038 1.0038 100.00 0.000 0.000 0.4 1.0150 1.0149 100.00 0.000 0.000 0.6 1.0331 1.0332 100.00 0.000 0.001 0.8 1.0580 1.0582 100.00 0.000 0.001 1 1.0892 1.0893 100.00 0.000 0.002 2 1.3178 1.3181 99.99 0.000 0.006 3 1.6257 1.6260 99.98 0.000 0.015 4 1.9760 1.9768 99.98 0.000 0.021 5 2.3501 2.3512 99.96 0.000 0.037 Table 3. Convergence of the fundamental frequency ratios for a simply supported square plate (Poisson’s ratio=0.3) Mesh sizes and (CD/Qjn at W^,Jh No. of modes and (co/coUn at W^Jh 4 modes 1.0 1.4 8x8 mesh 1.0 1.4 6x6 1.4174 1.7423 1 1.7028 7x7 1.4163 1.7396 2 1.4169 1,7433 8x8 1.4164 1.7403 4 1.4164 1.7403 9x9 1.4164 1.7400 5 1.4163 1.7401 Table 4. The lowest three frequency ratios and the modal participations for a simply supported square plate (Poisson’s ratio=0.3) Elliptic FEM integral[3] (CO/ffiiJii (to/oiOii Modal Participation % qu qi3 -1- qsi qi3 - qsi q33 qi5 -f. qsi 0.2 1.0195 1.0195 99.93 0.07 0.00 0.00 0.00 0.4 1.0757 1.0765 99.72 0.27 0.00 0.01 0.00 0.6 1.1625 1.1658 99.38 0.59 0.00 0.02 0.00 0.8 1.2734 1.2796 98.93 1.02 0.00 0.05 0.01 1.0 1.4024 1.4163 98.34 1.57 0.00 0.08 0.01 1.2 1.5448 1.5659 97.54 2.30 0.00 0.15 0.01 1.4 1,6933 1.7401 96.29 3.42 0.00 0.27 0.02 (C0/®l)21 q2i q23 q4i q43 ... 0.2 N/A 1.0243 99.93 0.06 0.01 0.00 ... 0.4 N/A 1.0976 99.50 0.45 0.03 0.01 — 0.6 N/A 1.2072 98.15 1.28 0.54 0.02 ... 0.8 N/A 1.3411 97.54 2.41 0.00 0.05 — 1.0 N/A 1.5126 96.24 3.69 0.00 0.08 ... 1.2 N/A 1.6900 94.90 4.92 0.03 0.15 ... 1.4 N/A 1.8952 93.54 6.01 0.01 0.44 ... y/raJh (C0/0)l)22 q22 q24 + q42 q24 - q42 q44 ... 0.2 N/A 1.0245 100.00 0.00 0.00 ... 0.4 N/A 1.0751 100.00 0.00 0.00 0.00 ... 0.6 N/A 1.1611 99.99 0.00 0.00 0.01 ... 0.8 N/A 1.2806 99.98 0.01 0.00 0.01 ... 1.0 N/A 1.4041 99.93 0.01 0.00 0.06 — 1.2 N/A 1.5551 99.97 0.01 0.00 0.01 ... 1.4 N/A 1.7074 99.98 0.02 0.00 0.00 — Table 5, Convergence of the fundamental frequency ratios for a clamped beam No. of elements and 4 modes (co/©l)i at W™/r No. of modes and (©/©lOi at Wn.ax/r 3.0 5.0 25 elements 3.0 5.0 10 1.1751 1.4046 1 15 1.1740 1.4009 2 20 1.1732 1.3999 3 25 1.1731 1.3996 4 1385 Table 6 The fundamental frequency ratios and the modal participations for a clamped beam Elliptic integral (0)/(0l)i FEM (ffl/(0L)l Modal Participation % qi q3 qs _ 1.0222 1.0222 99.78 0.20 0.02 0.00 2.0 1.0857 1.0841 99.33 0.58 0.08 0.02 3.0 1.1831 1.1731 98.35 1.44 0.17 0.04 4.0 1.3064 1.2817 97.37 2.28 0.29 0.07 5.0 1.4488 1.3996 96.26 3.22 0.42 0.11 Table 7 The fundamental frequency ratios and the modal participations for a simply supported rectangular (0/45/-45/90)s composite plate (a/b=2) WmJh Modal Participation % (ffl/coOn qu qi2 q2i qi3 q22 q23 qsi 0.2 1.0408 99.51 0.00 0.00 0.41 0.07 0.00 0.02 0.4 1.1490 96.57 0.00 0.00 3.01 0.24 0.00 0.17 0.6 1.3484 92.93 0.00 0.00 4.55 0.47 0.00 2.04 0.8 1.5241 98.51 0.00 0.00 0.53 0.94 0.00 0.02 1.0 1.7190 97.43 0.00 0.00 2.39 0.11 0.00 0.07 1.2 1.9258 95.78 0.00 0.00 3.57 0.62 0.00 0.02 1.4 2.1409 94.27 0.00 0.00 4.84 0.77 0.00 0.13 Table 8 The fundamental frequency ratios and the modal participations for a simply supported rectangular (0/90) composite plate W„,Jh Modal Participation % (CO/COiOn qii qi3 qsi q33 0.2 1.0358 97.83 1.18 0.82 0.18 0.4 1.1432 95.13 2.25 2.24 0.38 0.6 1.2993 94.53 3.86 1.18 0.60 0.8 1.5432 88.56 4.36 4.77 2.31 1.0 1.7880 89.15 3.31 5.06 2.48 1.2 2.0142 92.22 2.89 3.15 1.74 1.4 2.2823 92.01 2.92 2.92 2.15 1386 0.5 1 1.5 2 -6 -3 Time Ratio (tn‘1) 4 6 8 10 12 Frequency Ratio U Displacement Figure 1. Time histories, phase plot and PSD for the fundamental mode at =5.0 of a simply supported beam Power Spectrum Density Total OlsplacemonllThIckness qSSn'hJcknBSS q11/ThIckness p P P T* Thousandths o o o Time Ratio (tn"11) -0.4 -0.2 0 0.2 0.4 Displacement Figure 2a. Time histories, phase plot and PSD for the fundamental mode at W^u/h =1.0 of a simply supported square plate 1388 Thousandths Displacement Spectrum Density Total Displacement/Thickness q22/Thickness q1 IH'hickrtess 0 12 3 4 Time Ratio {t/T11) 0 12 3 4 0 2 4 6 8 10 12 Frequency Ratio 0 12 3 4 Time Ratio {tfTII) 0 12 3 4 Time Ratio (t/TII) Displacement Figure 4. Time histories, plot and PSD for the fundamental mode at Wmax/h =1.0 of a simply supported (0/45M5/90)s rectangular plate 1391 Power Specirum DonsUy Total Dlsplacemenin'hlckness qSin’hlckness q11/Thlckness NONLINEAR FORCED VIBRATION OF BEAMS BY THE HIERARCHICAL FINITE ELEMENT METHOD P. Ribeiro and M. Petyt Institute of Sound and Vibration Research, University of Southampton, Southampton S017 IBJ, UK Abstract: The hierarchical finite element (HFEM) and harmonic balance methods are used to derive the equations of motion of beams, in steady-state forced vibration with large amplitude displacements. These equations are solved by the Newton and continuation methods. The stability of the obtained solutions is investigated by studying the evolution of perturbations of the solutions. Additionally, a method that allows a quick examination of the stability of the solution is presented and applied. The convergence properties of the HFEM, the influence of the number of degrees of freedom and of in-plane displacements are discussed. The HFEM results are compared with experimental results. Symbolic computation is used in the derivation of the model. NOTATION A - extension coefficient b - width of the beam B - coupling coefficient [C] - damping matrix D - bending coefficient [D] - Jacobian of {F} |D| - determinant of [D] E - Young's modulus [E] - elastic matrix {f} - vector of out-of plane shape functions |F} - vec. of amplitudes of generalised forces - vector of generalised forces {F} - vector of dynamic forces {g} - vector of in-plane shape functions h - length of the finite elements h - thickness of the beam [Klb] - linear bending stiffiiess matrix [Kip] - linear stretching stiffiiess matrix [K2], [K3] and [K4] - components of nonlinear stiffiiess matrix [Knl] - nonlinear stiffiiess matrix L - length of the beam [M] - mass matrix [Mb] - bending mass matrix [Mp] - in-plane mass matrix [N] - matrix of shape functions [N'''(x)J - row matrix of out-of-pl. sh. f p j - number of in-plane shape functions pQ - number of out-of-plane s. funct. {qp} - in-plane displacement fimction {qw} - transverse displacement function r - radius of gyration t - time u - in-plane displacement Ur - generalised in-plane displacements w - transverse displacement Wr - generalised out-of-pl. displ. {wc}, (wj - coef of cosine and sine terms X - axial coordinate of the beam a - loss factor p - damping factor eg ,s§ - linear membrane and bending strains - geometrically nonlin. membrane strain {si}, {£ 2 }- linear and geom. nonl. strains 5 W - virtual work of the external forces cx SWy - virtual work of the internal forces 5W- - virtual work of the inertia forces in X - characteristic exponent p - mass density CO - angular fi:equency cOjj. - natural frequencies Ja>y - diagonal matrix of squares of natural firequencies - Viscous damping ratio 1393 1-INTRODUCTION In real systems, due to large amplitudes of the excitation, small stiffiiess or excitation with a frequency in the neighbourhood of resonance frequencies, vibrations with large amplitudes can occur. In this case, the linear theories may not allow a good representation of the dynamic characteristics of the system. A typical case study of vibrations in the nonlinear regime is the forced vibration of beams, with large displacements. Although a large amount of investigation has been carried out in this field [1, 2, 10, 18 and others], a method that would allow the inclusion of higher order mode contributions and damping, without increasing excessively the number of degrees of freedom (d.o.f) is desirable. The purpose of this work is to apply and investigate the advantages of a method that satisfies these conditions: the hierarchical finite element method (HFEM). In the HFEM, to achieve better approximations, higher order shape functions are added to the existing model. Convergence tends to be achieved with far fewer d.o.f than in the /z-version of the finite element method [4, 11]. The linear matrices possess the embedding property, meaning that the associated element matrices for a number of shape functions n=ni are always submatrices for n=n 2 , n 2 >ni. The existing nonlinear matrices of an approximation of lower order, ni, can be used in the derivation of the nonlinear, matrices of the improved approximation, n 2 . This makes the construction of a more accurate model, potentially quicker in the HFEM than in the /z-version. We are going to consider that the time variation of the solution may be expressed by harmonics and use the harmonic balance method (HBM). Compared with perturbation methods, the main advantages of the HBM are its simplicity, the fact that it is not restricted to weakly non-linear problems and, for smooth systems, the assurance of convergence to the exact solution [5]. In nonlinear vibrations, the frequency response curves can have multi-valued regions, turning and bifurcation points. In these regions, we are going to use a continuation method [8, 14], because, if the Newton method alone is applied, the solution will depend heavily on the initial guess and convergence is very difficult to achieve. Symbolic computation [15] will be utilised, allowing an easier and more accurate construction of the model. 1394 2 - MATHEMATICAL MODEL The beam is assumed to be elastic and isotropic, with thin uniform thickness h. The effects of transverse shear deformations and rotatory inertia are neglected. The transverse displacement is large compared with the beam thickness, but is very small compared with the length of the beam (w « L). The slopes are also very small; (w «1. The displacement components u and w may be expressed as the combination of the hierarchical polynomial shape functions, §lS2"-Spi ^ 0 f,f,..-fn {d} = |;;) = [N]{q}.[N] = {q}"=[U, ... U,, W. ... Wj. (2.1) The hierarchical polynomials used in this study were derived from Rodrigue's form of Legendre polynomials [4]. Only one element was used to model the whole beam and only the shape functions that satisfy the boundary conditions were included in the model. Applying the theory of Bemoulli-Euler, expressing the strain; 1 as {s} = (w^f /2 0 P _JOo 'Oj ■ ( 2 . 2 ) I 0 j L8 “j [oJ -- -- and equating the virtual work of the inertia forces (D’ Alembert principle) to the virtual work of the external and elastic restoring forces we obtain: {5q}^{F}-bJJ{5s,}%{5s,r)[E]({e,} + {s,})dL= {5q}^[M]{q}, (2.3) ° ; A. D = £(1,2^)Ed2; [M] = phbJjN]^[N]dL. [E] = The stiffness matrices are defined by: b[{58,}^[E]{s,}dL={5q}"[Kl]{q}.bJj6eT[E]{8,}dL={6qnK2]{q}. b({5e3r[E]{8,}dL={6q}"[K3]{q}, b[{5e,}"[E]{84dL={5qnK4]{q}. (2.4) Considering only transverse forces, if P(t) represents a concentrated force acting at the point x=Xj and (x, t) represents a distributed force, the generalised forces are {F} = (£p,(x.t){N”(x)}dL + P,(t){N”(x.)}). (2.5) In real systems energy is dissipated; consequently, damping should be included in the present model. For a large variety of materials experimental investigations show 1395 that the energy dissipated per cycle is not dependent on the frequency and is proportional to the square of the amplitude of vibration [12, 13], The corresponding type of damping is called hysteretic. We will represent it by a matrix proportional to the mass matrix and frequency dependent: [C] = i[M] (2.6) Considering that damping in the beam results only from the action of the linear axial and bending strains, we have the following equations of motion; "0 _K3 K4jj (2.7) The in-plane inertia can be neglected for slender beams [3] and the damping contribution due to the axial stress is generally negligible compared to that due to the bending stress [12], With these approximations and because [K3] = 2[K2]^, ref [4], we can simplify the equations of motion to obtain: +J[M.]{qw} + [Kl.]{q„} + [Knl]{q„} = {F}, (2.8) [Knl] = [K4] - 2[K2 ]^[k1,]'‘[K2] . (2.9) To integrate exactly terms involving shape functions or its derivatives, present in the stiffriess and mass matrices, symbolic computation was employed, using the package Maple [15]. If the external excitation is harmonic and if initial conditions are such that no transient response exists, then {qw(t)} may be expressed, in a first approximation, as: {qw(t)}={wc}cos(©t)+{Ws}sin(a)t) (2.10) We are going to insert this equation into the equations of motion (2.8) and apply the HBM. This method can be easily implemented in a program written with the symbolic manipulator For that, one defines the command trign using Maple library of trigonometric functions, trig, in the following way; trign: =readlih('trig/reduce') [15, 17]. trign, thus defined, replaces all nonlinear trigonometric functions by linear ones^. With the command coeff one selects the terms in cos(ot) and sin(a 3 t). In this way, we obtain equations of motion of the form: 1 For example cos^(Q)t) is replaced by ^cos(a)t) + .icos(3cot) 1396 -CO' M, 0 0 M, -3M, KL ( 2 . 11 ) {Fj=|0Knl]{q4cos(cDt)dt=gKM.l + iKM.3){wJ+iK^^^ (2.12) {F,}=||;[KM]{q4smMdt =iKNL2{wJ + gKNLl + ^K^o]{w^ (2.13) where KNLl is a iunction of {Wc} only, KNL2 is a function ofboth (wc) and (wj and ICNL3 is a function of {Ws} only These three matrices are, as well as Mb and Klb, symmetric. 3 - STUDY OF THE STABILITY OF THE SOLUTIONS We will study the problem of local stability of the harmonic solution by adding a small disturbance to the steady state solution {q} = {q„} + {8qw} (3.1) and studying how the variation of the solution evolves. If {6q^} dies out with time then {q^} is stable, if it grows then {q^} is unstable. Inserting the disturbed solution (3.1) into equation (2.8), expanding the nonlinear terms into Taylor series around {q„} and ignoring terms of order higher than {bq^j, we obtain the variational equation: K]{5q„} + ^K]{6q„} + [KlJ{5q4 + ^Mj^{5q.}=M (3-2) The coefficients are periodic functions of time. With symbolic manipulation, they can easily be expanded in a Fourier series. If {q^} is of the form (2.10) and since is quadratic in {q„}, we have: 3{qw} ^(M|^=[[p.] + [p,]cos(2cst) + [p,]sin(2o)t)]. (3.3) 2 With this formulation, KNL2 must be calculated using 2[N* J{wJ[N*J(w,} , otherwise .iKNL2 should be considered instead of iKNL2 ■ 1397 Simplification (3.14) was possible because the damping matrix is, after transformation into modal coordinates, equal to a scalar matrix. Now, following Hayashi [7, page 93], we will express the solution of (3.14) in the form: |6f| = e^({bi)cos(®t) + {aj}sin(®t)) (3.15) which should allow us to determine, in a first approximation, the first order simple unstable region. Inserting (3.15) into (3.14) and appl 3 dng the HBM, we find (X^[l] + X[M,] + [M,]){y = {°} (3.16) 0 2cd[I] [-2cd[I] 0 J [Mo] = where [M,] = (3.17) ^ "1 p'" [BriDjB] [Bf[D„lB]-L^+ril]1[l]^[a>o/] (3.18) To determine the characteristic exponents, X, we transform this system into [16] 0 [I] Ifxl -[M„] -[M,]JlrJ (3.19) where {X} is a vector formed by {bi}, {aj. The values of X are the eigenvalues of the double size matrix in the previous equation. Bearing in mind that it is the stability of the variable {5^} in which we are interested we substitute equation (3.15) into equation (3.13) to obtain {61} = {{b,} cos( 0 t) + {a, }sin((Dt)}. (3.20) If the real part of X^ - ii 2 © is positive for any then the solution is unstable, otherwise it is stable. 1398 For undamped systems, it was demonstrated in [8], that important conclusions about the stability of the solutions can be deducted from the determinant of the Jacobian of {F}. We are going to extend the demonstration to systems with mass proportional damping. Applying the derivation rule for composite functions, we obtain the derivatives of {Fj} and {fJ with respect to {wc} and {wj as follows; Matrices [I] and [Mo] are symmetric and matrix [Mi] is skew-symmetric. This means that the eigenvalues of equation (3.16) are either purely imaginary or purely real [8]. If X is imaginary the solution is always stable; if X is real the stability limit is defined by = (3.25) Inserting (3.25) in (3.16) we arrive at ■[Bf[D„][B]-co^[l] + [o.,/] [Bf[D.][B] + P l|b,l Jol [Bf[D„][B]-p [BnD,,][B]-<B^[l] + [co,/]JlaJ loj The matrix in the previous equation is [B]^[D][B], where [D] is the Jacobian of {F’} with respect to the vector of coefficients of the cosine and sine terms, given by [D] = d{F}/d A non-trivial solution of (3.26) exists if det([Bf [D][B])=0 o |BnDl=0o|Dl=0. (3.28) The last equivalence is true, because [B] is a non-singular matrix. Thus, we proved that in the stability limit, the determinant of the Jacobian of (F), |D| , is zero. |D| is a polynomial in the coefficients {wj and (wj and in o; therefore, it is a continuous function in those coefficients. All the experimental and numerical analysis of nonlinear vibration of beams, indicate that the shape of vibration, defined in our model by {wc} and {w^}, is a continuous function of the amplitude and the frequency of vibration. Thus, we conclude that |d| varies in a continuous way through the FRF (fi-equency response function) curve. Hence, if there is a change in its sign between two consecutive points of the FRF curve, then |D| =0 for a particular point between these two. In that particular point, the stability limit might have been crossed. So, a complete study of the first order solution’s stability may be carried out by determining only the characteristic exponents of the first solution and when JD] changes sign or when |D| is approximately zero. As jD| is needed in the continuation method and, when the Newton method is applied, can be easily calculated from [D], this results in substantial time savings. 4-APPLICATIONS A clamped-clamped beam made in an aluminium alloy with the reference 7075- T6 was analysed. Its material [9] and geometric properties are: E = 7.172*10^° N/m2, p = 2800 kg/m^, h=0.002 m, b=0.02 m, L = 0.405 m 3. For aluminium, a typical value of the loss factor (which is multiplied by the stiffness matrix) is a=0.01 (C=0.5%), but the measured value in reference [18] was approximately equal to 0.038 (^sl.9%)^. In order to have the same damping coefficient for the first mode of vibration, the value of the damping factor is: P = (»o,^xa. (4.1) The beam was modelled using the HFEM, as described in section 2. To solve the system of equations (2.11), Newton’s method was used in the nonresonant area. In the vicinity of resonance frequencies it is difficult to obtain convergence by the Newton method 3 Except in the comparison with experimental results, where L-0.406 (value of Wolfe’s clamped- clamped beam length). Wolfe did not think that the measured damping ratio was only due to material damping. He also attributed the obtained value to damping in the joints and to the coil magnet arrangement used to excite the beam. 1400 and a continuation method was applied [8, 14]. The derivation of the Jacobian matrix present in both methods was performed symbolically [15]. Because the excitation force will be applied in the middle of the beam and both the beam and the boundary conditions have symmetric properties, only symmetric out of-plane shape functions and antisymmetric in-plane shape functions will be used^. 4.1 - Study of convergence with number of shape functions With four out-of-plane (po) and four in-plane shape functions, convergence of the value of the first linear natural frequency is achieved (Table 1). This number of shape functions will be the starting value for our nonlinear analysis. The number of degrees of freedom of the present damped model is equal to 2xpo. Table 1 - Natural linear frequencies of the cc beam (rad/s). Mode 1. Exact Po=2, pi=2 Po=3, pi=3 Po=4, p=4i Po=5, Pi=5 396.6 396.613 239 396. 605 011 396. 605 008 396. 605 008 In Figures 1, 2 and 3 we can see the FRFs in the vicinity of the first, third and fifth mode, obtained when a force P of 0.03 N was applied. Near the first mode there is no increase in accuracy by using more than four out-of-plane and four in-plane shape functions (po=4, pi=4). However, for the third mode, as the amplitude of vibration grows, the results obtained with po=4 and pi=4 depart from the ones obtained with more shape functions. The FRF curve constructed with po=5 and pi=5 is quite similar to the coincident FRFs obtained with po=6, pi=6 and with po=7, pi=7. In the neighbourhood of the fifth mode, convergence seems to be achieved with po=8 and pi=8. 4.2 - Influence of in-plane displacements In Figure 4 the FRFs obtained considering and neglecting the in-plane displacements are compared. As in references [1] and [10], we found that the in-plane displacements ‘reduce’ the non-linearity, in the sense that the non-linearity caused by them is of the soft spring type and counterbalances the hard spring type non-lineanty 5 To check if the nonlinearity introduced any coupling and consequent antisymmetric terms in the response, a model including symmetric and antisymmetric, in- and out-of-plane shape functions was considered. It was confirmed that, with these boundary conditions and with the one harmonic representation of the solution’s time dependency, there is no such coupling. 1401 caused by the transverse displacements. This ‘reduction’ of nonlinearity is due, as the formulation of the nonlinear stiffness matrix - eq. (2.9) - shows, to the effects of in¬ plane deformation on the stiffness of the structure. 4.3 - Study of stability In Figures 5 and 6 we can see the stability studies carried out in the neighbourhood of the first and third resonance frequencies, using po=6, pi=6 shape functions and with an excitation force of amplitude P = 0.03 N. In all cases, |D| changed sign when the stability condition of the solutions changed. 4.4 - Comparison with experimental results In Figures 7 and 8 we can see the comparison between the FRF obtained with the HFEM, using po=6, pj=6 shape functions, and the experimental results [18] when a force P = 0.134 N is applied in the centre of the beam. Two values were used for the damping factor: P=0.01 g3q, and P=0.038 g)o, . The HFEM provides a FRF with a slope similar to the experimental one around the resonance frequency. This indicates that the nonlinear stiffness is well represented by the model. The turning point corresponding to the largest amplitude of vibration, where the jump phenomena occurs, obtained with the HFEM, point B, does not match the experimental one, point A. With the typical value used for the loss factor in aluminium alloys, a=0.01 (p=0.01oi)oj), the maximum amplitude of vibration was more than double the one measured. However, the HFEM solutions represented in Figure 7 after point A are very close. Thus, in a real system, a small perturbation would easily make the shape of vibration change into an unstable one and a change, or jump, to a stable shape of vibration at a lower amplitude could be observed before the largest computed amplitude of vibration was achieved. With the measured loss factor, a=0.038, the largest amplitude of vibration obtained with the HFEM, was around a half of the measured maximum amplitude. 1402 5 - CONCLUSIONS The HFEM dynamic model of a beam vibrating with large amplitudes was constructed with small time expense. This is due to the small number of degrees of freedom with which convergence is achieved, to the easy way in which the number of d.o.f are reduced, benefiting from the symmetry properties of the problem, and to the embedding properties^ of the HFEM. For the amplitudes of vibration displacement studied, with relatively few d.o.f the FRF curves were accurately determined until the 5th order mode, inclusive. If modes of order higher than 5th are to be studied, then the inclusion of more elements instead of more shape functions should be considered, as shape functions of excessive high order turn the construction of the matrices quite time consuming. The comparison with experimental results showed a very good prediction of the slope of the FRF by the HFEM. The largest amplitude of vibration and the correspondent turning and jump point, are greatly influenced by the amount of damping used. Using the flexibility of choosing the shape functions in the HFEM model it was shown that the in-plane displacements cause a softening-type nonlinearity. With the continuation method the multi-valued regions of the FRF curves were completely and automatically described. To determine the characteristic exponents that establish the stability of the solution, we solved an eigenvalue problem. Due to the reduced number of degrees of freedom of the HFEM model this was quickly solved. More important, it was proven that in the stability limit the determinant |D| is zero. Thus, we only have to determine the characteristic exponents of the first solution and when there is an indication that |Dj =0 for a particular point, to check if the stability of the solution changed. This results in significant time savings. With symbolic computation, the matrices involved in the HFEM model and the Jacobian matrix necessary in the continuation and Newton methods, were easily and exactly derived, thus reducing the numerical errors. Symbolic computation was also helpful in the application of the HBM. ^ Here we include the HFEM’s advantages in the derivation of the nonlinear stiffness matrix. 1403 REFERENCES [ 1 ] - Atluri, S., Nonlinear vibrations of a hinged beam including nonlinear effects. Trans, of the ASMS J. of Apl. Mech., 1973, 40, 121-126. [ 2 ] - Bermet, J. A. and Eisley, J. G., A multiple-degree-of freedom approach to nonlinear beam vibrations. J. of the Am. Inst, of Aeronaut, and Astronaut., 1970,8, 734-739. [ 3 ] - Cheung, Y. K. and Lau, S. L., Incremental time-space finite strip method for non-linear structural vibrations. Earthquake Engng. and Struct. Dynamics, 1982,10,239-253. [ 4 ] - Han, W, The Analysis of isotropic and laminated rectangular plates including geometrical non-linearity using the p-version finite element method^ Ph.D. Thesis, University of Southampton, Southampton, 1993. [ 5 ] - Hamdan, M.N. and Burton, T.D., On the steady state response and stability of non¬ linear oscillators using harmonic balance. J. of Sound and Vibr., 1993,166, 255-266. [ 6 ]- Stokey, W. F., Shock and Vibration Handbook, Third edition, ed. C. M. Harris, McGraw-Hill, New York, 1988, p. 7-14. [ 7 ] - Hayashi, C, Nonlinear Oscillations in Physical Systems, McGraw-Hill, New York, 1964. [ 8] - Lewandowski, R., Non-linear, steady-state analysis of multispan beams by the finite element method. Computers and Struct., 1991,39, 83-93. [ 9 ] - ASM Committee on Properties of Aluminium Alloys, Properties and Selection of Metals, Metals Handbook, Vol. 1, 8th edition, ed. T. Lyman, Ohio, 1961, p. 948. [ 10 ] - Mei, C and Decha-Umphai, A finite element method for non-linear forced vibrations of beams. J. of Sound and Vibr., 1985,102, 369-380. [ 11 ] - Meirovitch, L., Elements of Vibration Analysis, McGraw-Hill, Singapore, 1986. [ 12 ] - Mentel, T. J., Vibrational energy dissipation at structural support junctions. In Colloquium on Struct. Damping, ed. E. J. Ruzicka, 1959, pp. 89-116. [ 13 ] - Petyt, M, Introduction to Finite Element Vibration Analysis, Cambridge University Press, Cambridge, 1990. [ 14 ] - Ribeiro, P. and Petyt, M., Study of nonlinear free vibration of beams by the hierarchical finite element method. ISVR Techmcal Memorandum No.773, University of Southampton, Southampton, November 1995. [ 15 ] - Redfem, Darren, The Maple Handbook, Springer-Verlag, New York, 1994. [ 16 ] - Takahashi, K., A method of stability analysis for non-linear vibration of beams. J. of Sound and Vibr., 1979, 67, 43-54. [ 17 ] - Wang, S. S. and Huseyn, K., Bifurcations and stability properties of nonlinear systems with symbolic software. Math. Comput. Modelling, 1993,18, 21-38. [ 18 ] - Wolfe, Howard, An experimental investigation of nonlinear behaviour of beams and plates excited to high levels of dynamic response, Ph.D. Thesis, University of Southampton, Southampton, 1995. 1404 FIGURES mnxiwi 1 h 0.5- €f^ ().4- 0.3- . g 0.2- « O.I- . • O' 340 360 380 4^0 420 0 ) (rad/s) 1 - FRF in the vicinity of the first mode of vibration. x=0.5 0 po=4, pi=4; □ po=5, pi=5; 0 po=6, pi=6; + po=7, pi=7. 1405 innxlw h 1 0.5- 9 °’ 0.4- t , . 0.3- 0.2 % 0.1 . . - * * ‘ . 0^ 340 360 380 400 420 440 CO (rad/s) Figure 4 - FRF with in-plane displacements, po=6, Pi=6 (o), and without in-plane displacements, po=6, Pi=0 (+). x=05xL. Figure 5 - Stability study. First mode. x=0.5xL. □ stable solution; + unstable solution; po=6, pi=6. 1406 mnxiwi h B 2.2- 2- 1.8- 1.6- 1.4- 1,2- I- A 0.8 0.6- 0.4' 0.8 1 .2 , 1.4 1.6 1.8 CO/OD„ 0 Figure 7 ~ Comparison with experimental results, o HFEM stable, □ HFEM unstable, Po=6 and pi=6, |3=0,01cOo,2; + experimental. x=05xL. mnxiw h 1- A 0.8- * 0.6 B 0.4 O. 2 J . • * 0.8 0.9 1 1.1 M/Wq 1.2 1.3 Figure 8 - Comparison with experimental results, o HFEM stable, □ HFEM unstable, po=6 and pi=6, ( 5 = 0 . 038 cOo, 2 ; + experimental. x=05xL. 1407 1408 GEOMETRICALLY NONLINEAR DYNAMIC ANALYSIS OF 3-D BEAM Kuo Mo Hsiao and Wen Yi Lin Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan, Republic of China ABSTRACT A co-rotational finite element formulation for the geometrically nonlinear dynamic analysis of spatial beam with large rotations but small strain is presented. The deformation nodal forces and inertia nodal forces are derived by using the d'Alembert principle and the virtual work principle. The gyroscopic effect is considered here. The beam element developed here has two nodes with sbc degrees of freedom per node. Some angular velocity coupling terms, which are so called gyroscopic forces, are obtained in inertia nodal force. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the nonlinear dynamic equilibrium equations. Numerical examples are presented to demonstrate the acctuacy and efficiency of the proposed method. INTRODUCTION In recent years, the nonlinear dynamic behavior of beam structures, e.g., framed structures, flexible mechanisms, and robot aims, has been the subject of considerable research. In [1], Hsiao ^d Jang presented a co-rotational formulation and numerical procedure for the dynamic analysis of planar beam structures. This formulation and numerical procedure were proven to be very effective by numerical examples studied in [1]. However, it is only limited for planai* beam structures. A general formulation for three dimensional beam element is not a simple extension of a two dimensional formulation, because large rotations in three dimensional analysis are not vector quantities; that is, they do not comply with the rules of vector operations. In [2] a motion process of the three dimensional beam element is proposed for the large displacement and rotation analysis of spatial frames. In [3] a co- rotational formulation for three-dimensional beam element is proposed. However, it is only limited for nonlineai' static analysis. The objective of this study is to present a practical formulation for the dynamic analysis of three dimensional Euler beam. The kinematics of the beam element proposed in [3] is adopted here. 1409 The element deformations are determined by the rotation of element cross section coordinates, which are rigidly tied to element CTOSS section, relative to the element coordinate system [2, 3]. The three rotation parameters proposed in [3] are used to determine the orientation of the element cross section coordinates. In order to capture the gyroscopic effect, the relation between the time derivatives of the rotation parameters and the angular velocity and the angular acceleration is derived here. The beam element developed here has two nodes with six degrees of freedom per node. The element nodal forces are conventional forces and moment. The deformation nodal forces and inertia nodal forces are derived by using the d'Alembert principle and the virtual work principle in the current element coordinates. An incremental-iterative method based on the Newmark direct integration method and the Newton- Raphson method is employed here for the solution of the nonlinear dynamic equilibrium equations. Numerical examples are presented to demonstrate effectiveness of the proposed method. FINITE ELEMENT FORMULATION Basic assumptions The following assumptions are made in the derivation of the nonlinear behavior: (1) the beam is prismatic and slender, and the Euler-Bernoulli hypothesis is valid; (2) the centroid and the shear center of the ctoss section coincide; (3) the unit extension and twist rate of the centroid axis of the beam element are uniform; (4) the cross section of the beam element does not deform in its own plane, and strains within this cross section can be neglected; (5) the out-of¬ plane warping of the cross section is the product of the twist rate of the beam element and the Saint Venant warping function for a prismatic beam of the same cross section; (6) the deformations of the beam element are small. Coordinate systems In this paper, a co-rotational total Lagrangian formulation is adopted. In order to describe the system, following [3], we define three sets of coordinate systems (see Fig. 1): (1) A fixed global set of coordinates, Xj(z = 1,2,3); the nodal coordinates, displacements, and rotations, and the stiffness matrix of the system are defined in this coordinates. (2) Element cross section coordinates, rf (f = 1,2,3); a set of element cross section coordiaates is associated with each cross section of the beam element. The origin of this coordinate system is rigidly tied to 1410 the shear center of the cross section. The xf axes are chosen to coincide with the normal of the corresponding cross section and the X 2 and X 3 axes are chosen to be the principal directions of the cross section. (3) Element coordinates, x, (z = 1,2,3); a set of element coordinates associated with each element. The origin of this coordinate system is located at node 1 ; the xj axis is chosen to pass through two end nodes of the element, and the X 2 and X 2 axes are determined from the orientation of the element cross section coordinates at two end nodes using the way given in [2]. The deformations and stiffness matrices of the elements are defined in terms of this coordinates. In this paper the element deformations are determined by the rotation of element cross section coordinates relative to this coordinate system. Rotation vector and rotation parameters For convenience of the later discussion, the term 'rotation vector' is used to represent a finite rotation. Figure 2 shows that a vector b which as a result of the application of a rotation vector (pa. is transported to the new position h'. The relation between b and b' may be expressed as [4] b' = cos 0 b + (1 - cos 0 )(a • b) + sin 0 (a x b), ( 1 ) where (p is the angle of counterclockwise rotation, and a is the unit vector along tiie axis of rotation. In this paper, the s)mbol { } denotes column matrix. Let e,- and ef (i - 1 , 2 , 3 ) denote the unit vectors associated with the x, and xf axes, respectively. Here, the traid ef in the deformed state is assumed to be achieved by the application of the following two rotation vectors to the traid Cj : e„ = 0 „n, = 0 it, where n={0, 82! (el + 03/(0! + 0|)V2} = {0,^2,713}, t = {cos 0 „, 62 , 63 }. cos 0 „=(i- 0 !-e!)Vl ^ dw{s) . dv(s) (2,3) ( 4 ) 1411 in which n is the unit vector perpendicular to the vectors ei and ef, and t is the tangent unit vector of the deformed centroid axis. Note that ef coincides with t. is the inverse of cosd„. v(s) and w(s) are the lateral deflections of the centroid axis of the beam element in the X 2 and directions, respectively, and s is the arc length of the deformed centroid axis. The rotation vectors e„ and 0^ are determined by (f = l,2,3). Thus, di are called rotation parameters in this study. Using Eqs. (l)-(4), the relation between the vectors and ef (i = 1,2,3) in the element coordinate system may be obtained as ef = [t, Ri, R2] = Re,-, Rl = cos^iri + sin0ir2, R 2 = -sin^iri + cos0ir2, ri = {-03, cos 0„ + (1 - cos 0„, (1 - cos d„)n 2 n^h r2 ={02^(1-cos0„)w2W3,cos 0„ +(l-cos0„)n3}, (5) where R is the so-called rotation matrix. Let 0 = {01, 02, 03> be the vector of rotation parameters, 36 be the variation of 0. The traid ef corresponding to 0 may be rotated by a rotation vector = {3ipi, 3(l>2, <5^3} to reach their new positions corresponding to 0 + 50 [3]. When 02 and 03 are much smaller than unity, the relationship between 50 and 5<^ may be approximated by r 1 50 = -03 ^2 03/2 -02/2' 1 0 0 1 ( 6 ) If both sides of Eq. (6) is divided by St, the first time derivative of 0 may be expressed by r 1 0 = -03 02 03/2 - 02 / 2 - 1 0 0 1 (j) = T^^ij), (7) where the symbol ( ) denotes differentiation with respect to time t . 1412 = 1,2,3) denote the angular velocities about the axes. From Eq. (7), the second time derivative of 6 may be expressed by may be expressed as 9 = t-^(j) + T"^^, (8) where = 1,2,3) denote the angular accelerations about the Xi axes. Nodal parameters and forces The global nodal parameters for the system of equations corresponding to the element local nodes j (j - 1, 2) are ll^j, the Xj (i = 1,2,3) components of the translation vectors at nodes j, and the Xi (f = 1,2,3) components of the rotation vectors at nodes j. Here, the values of are reset to zero at current configuration. Thus, <50^, the variations of ^ij, represent infinitesimal rotations about the Xi axes [3], <i>ij and Oy represent angular velocities and angular accelerations about the Xj axes, respectively. Tlie generalized nodal forces corresponding to dOy are the conventional moments about the Xf axes. The generalized nodal forces corresponding to dllij, the variation of Uy-, are the forces in the X, directions. The element employed here has six degrees of freedom per node. Two sets of element nodal parameters termed 'explicit nodal parameters' and 'implicit nodal parameters' are employed. The explicit nodal parameters of the element are used for the assembly of the system equations from the element equations. Thus, they should be consistent with the global nodal parameters, and are chosen to be Uij, the x, (i = l,2,3) components, of the translation vectors uj at nodes j (j = 1, 2) and 0y, the Xi (i = 1,2,3) components of the rotation vectors (j)j at nodes j. Similarly, the generalized nodal forces corresponding to Wy and d^ij are /y and my, the forces in the Xi directions and the conventional moments about the Xj axes, respectively. The implicit nodal parameters of the element are used to determine the deformation of the beam element. They are chosen to be My, the Xj (i = l,2,3) components of the translation vectors u^ at nodes j and 0y, the nodal values of the rotation parameters 0- 1413 (i = 1,2,3) at nodes ; (j = 1, 2). The generalized nodal forces cori'esponding to duij and dOij are and m|, the forces in the Xi directions and the generalized moments, respectively. Note that are not conventional moments, because S 6 ij are not infinitesimal rotations about the Xi axes. In view of Eq. (6), the relations between the variation of the implicit and explicit nodal parameters may be expressed as dui' 1 0 0 0 dui dQi 0 TI^ 0 0 5<i>i ' dU2 ► — 0 0 I 0 5u2 502 0 0 0 Ti\ ^^2 (9) where daj = {duij,du 2 j,Su 3 j}, d 6 j={ddij, 5 d 2 j, 862 j}, and % ={ 8 <hj’^<p 2 j>^<p 3 jh (j = % 2). I and 0 are the identity and zero matrices of order 3x3, respectively. (j = 1 , 2) are nodal values of T"l Let f = {fi,mi,f2,m2K f® ={fi,mf,f2,m^}, where ij ={fij,f2jj3jh and mj (/’ = h 2), denote the internal nodal force vectors corresponding to the variation of the explicit and implicit nodal parameters, dq and, <5q^, respectively. Using the contragradient law [5] and Eq. (7), the relation between f and, f ^ may be given by f = (10) Kinematics of beam element The deformations of the beam element are described in the current element coordinate system. From the kinematic assumptions made in this paper, the deformations of the beam element may be determined by the displacements of the centroid axis of the beam element, orientation of the cross section (element cross section coordinates), and the out-of-plane warping of the cross section [3]. Let Q (Fig. 1) be an arbitrary point in the beam element, and P be the point corresponding to Q on the centroid axis. The position vector of point Q undeformed and deformed configurations may be expressed as 1414 and r^xei + ije2 + ze^, ( 11 ) r = + ^(s)e2 + ^(s)e3 + yel + ze| + Si^s^Qv (1^) where Xf,{s), v{s) and w{s) are the xy X 2 and X't, coordinates of point P, respectively, s is the arc length of the deformed centroid axis measured from node 1 to point P. ^c(®) expressed by Xc{s) = Mil + where un is the displacement of node 1 in the xi direction, and cos0„ is defined in Eq. (4). Here, z;(s) and w{s) in Eq. (12) are assumed to be the Hermitian polynomials of s , and 0i(s) in Eq. (12) is assumed to be linear polynomials of s, and maybe given by z;(s) = {Ni,N2 ,N3,N4}^{W21/%1/W22/%2}-= w{s) = {N'i,-N2,N3-,N4}^{m31,021/^32^^22} = 0l(s) = {N5,N6}^{0n^%> = N^u^, (14) Ni = 7(1 - ?)^(2 f I), Nj = |(1 -1^)(1 -1). 4 o yvj = i(i +1)2(2- f), Ni = |(-1 + ?2)(1 +1). 4 o Af5 = |(l-?). W6=|(l + |), (15) where S is the arc length of the centroid axis of the beam element and may be expressed by S^ltjQOS (17) where I is the chord length of the centroid axis of the beam element, and cos is given in Eq. (4). The way to determine the current element cross section 1415 coordinates at both ends, element coordinates, and element implicit nodal parameters corresponding to displacement increments is given in [ 2 ,3]. If ;c, y, and z in Eq, (11) are regarded as the Lagrangian coordinates, the Green strains en, Si 2 , and 613 are given by £11 = 1 2 \dxj [dxj -1 £12 2[^xJ 2[^xJ \dzj' (18) Substituting Eqs. (4), (5), (12), and (13) into Eq. (18), sn, ei 2 , and £13 can be calculated. Element nodal force vector The element nodal force vector (Eq. (10)) corresponding to the implicit nodal parameters are obtained from the d'Alembert principle and the virtual work principle. For convenience, the implicit nodal parameters are divided into four generalized nodal displacement vectors u,- (i = a,b,c,d), where Urt «{«!!, «i2}/ and u^, Uc, and u^ are defined in Eq. (14).. The generalized force vectors corresponding to (5Uj, the variation of u,- (f = a,h,c,d) are 4 - + 4 = where f^ and f*- {i = a,b,c,d) are the deformation nodal force vector and the inertia nodal force vector, respectively. The virtual work principle requires that 1416 du^Ja + < 50^4 + dn^ic + =JL {oiideii + 2 ai 2 dei 2 + 2 ai 3 ( 5 ei 3 + p8r^r)dV, where on = Esiy O 12 = 2Gei2 and 0-13 = 2Gei2, where E is tl^ Young's modulus and G is shear modulus, p is the density, and V is the volume of the undeformed beam. If the element size is properly chosen, the values of the nodal parameters (displacements and rotations) of the element defined in the current element coordinate system, which are the total deformational displacements and rotations, may always be much smaller than unity. Thus only the first order terms of nodal parameters are retained in deformation nodal forces. However, in order to include the effect of axial force on the lateral forces, a second order term of nodal parameters is retained. Because the values of the nodal parameters of the element may always be much smaller than unity, it is reasonable to assume that the coupling between the nodal parameters and their time derivatives are negligible. Thus only zeroth order terms of nodal parameters are retained in inertia nodal forces. From Eqs. (4), (5), and (12)-(21), the deformation nodal forces and the inertia nodal forces may be expressed as Li (22,23) . . rd Gj{di2-Qn)f 111 f“ = + k^)Uc, td- ^ i (24, 25) fa = (26) 4 = + m^)Ub - 2p4 , (27) fc = - 2ply fN'cdidsds , (28) f^ = m^Urf - p{ly - (29) 1417 where A is title cross section area, L is the initial length of the beam element, k,- and k^- {i = b,c) are bending and geometric stiffness matrices of conventional beam element [5,6], and J is the torsional constant p is the density, ly and 7^ are the moment of inertia of the beam cross section about the and axes respectively, m^ is the consistent mass matrix of bar element for axial translation, nij-f and m,> {i^b,c) are the consistent mass matrices of elementary beam element for lateral translation and rotation, respectively, and is the consistent mass matrix of bar element for axial rotation. These mass matrices can be found in [5, 6]. The underlined terms in Eqs. (27)-(29) are inertia forces induced by the gyroscopic effect, and are called gyroscopic forces. Element Matrices The element stiffness matrices and mass matrices may be obtained by differentiating the element nodal force vectors with respect to nodal parameters, and time derivatives of nodal parameters. However, element matrices are used only to obtain predictors and correctors for incremental solutions of nonlinear equations in this study. Thus, approximate element matrices can meet these requirements. The stiffness matrices and mass matrices of elementary beam element given in [5, 6] are also used here. Equations of motion The nonlinear equations of motion may be expressed by F^ = F^ + F^-P = 0 (30) where F^ is the unbalanced force among the inertia nodal force F^, deformation nodal force F^, and the external nodal force P. F^ and F^ are assembled from the element nodal force vectors in Eq. (10), which are calculated using Eqs. (10) and (22)-(29) first in the current element coordinate system, and then transformed from current element coordinate system to global coordinate system before assemblage using standard procedure. APPLICATIONS An incremental iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the nonlinear d 5 mamic equilibrium equations. 1418 The example considered is a right-angle cantilever beam subjected to an out-of-plane concentrated load as shown in Fig. 3. Four elements are used for discretization. A time step size of At = 0.25^ is used. The cantilever undergoes a finite free vibration with combined bending and torsion after the removal of the applied load; the time histories of out-of-plane displacements of the elbow and of the tip are given in Figs. 4 and 5. It is seen that the present results are in excellent agreement with those given in [7] and [8]. However, it should be mentioned that the beam elements used in [7] and [8] are derived using fully nonlinear beam theory and total Lagrangian formulation. Thus, the beam elements used in [7] and [8] are much more complicated than that proposed here. CONCLUSIONS A co-rotationai finite element formulation for the geometrically nonlinear dynamic analysis of spatial beam with large rotatio^ but small strain is presented. The deformation nodal forces and inertia nodal forces are derived by using the d'Alembert principle and the virtual work principle. The gyroscopic effect are considered here. The nodal coordinates, displacements, rotations, velocities, accelerations, and the equation of motion of the system are defined in a fixed global set of coordinates. The beam element developed here has two nodes with six degi'ees of fi^eedom per node. The element nodal forces are conventional forces and moments. AU of element deformations and element equations are defined in terms of element coordinates which are constructed at the current configuration of the beam element. The element deformations are determined by the rotation of element cross section coordinates, which are rigidly tied to element cross section, relative to the element coordinate system. In conjunction with the co-rotational formulation, the higher order terms of nodal parameters in element nodal forces are consistently neglected. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the nonlinear dynamic equilibrium equations. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method It is believed that the co-rotational formulation for 3-D beam element presented here may represent a valuable engineering tool for the d 5 mamic analysis of spatial beam structures. ACKNOWLEDGMENT The research was sponsored by the National Science Council, Republic of China, under contract NSC86-2212-E-009-006. 1419 REFERENCES 1. Hsiao, KM. and Jang, J.Y., . Nonlinear dynamic analysis of elastic frames. Computers & Structures, 1991, 33, 769-781. 2. Hsiao, KM. and Tsay, C.M., A motion process for large displacement analysis of spatial frames. International Journal of Space Structures, 1991, 6, 133-139. 3. Hsiao, K.M., Corotational total Lagrangian formulation for three-dimensional beam element. AIAA Journal, 1992, 30, 797- 804. 4. Goldstein, H., Classical Mechanics, Addision-Wesley, Reading, MA,1980. 5. Dawe, D.J., Matrix and Finite Element Displacement Analysis of Structures, Oxford Univ. Press, New York, 1984. 6. Hsiao, I<.M., A study on the dynamic response of spatial beam structures. NSC 82-0401-E009-081 Report, National Science Council, Taiwan, Republic of China,, 1993. 7. Simo, J.C. and Vu-Quoc, L., On the dynamics in space of rods undergoing large motions - A geometrical exact approach. Computer Methods in Applied Mechanics and Engineering, 1988, 66,125-161. 8. lura, M. and Atluri, S.N., Dynamic analysis of finitely stretched and rotated three-dimensional space-curved beams. Computers & Structures, 1988, 29, 875-889. Fig. 1 Coordinate systems. 1420 Fig.2 Rotational vector Material Properties: EA==10" EIy = EI^=GJ = 10^ Ap = 1 ply=pl,= 10 Time History of Load: Fig.3 Right-angle Cantilever beam 0 5 10 15 20 25 30 Time Fig. 4 Displacements in the X 3 direction at point B. Fig. 5 Displacements in the X 3 direction at point A. 1422 Nonlinear Response of Composite Plates to Harmonic Excitation Using The Finite Element Time Domain Modal Method Raymond Y, Y. Lee, Yucheng Shi and Chuh Mei Department of Aerospace Engineering Old Dominion University, Norfolk, VA 23529-0247 Abstract A multimode time domain formulation based on the finite element method for large amplitude vibrations of thin composite plates subjected to a combined harmonic excitation and thermal load is presented. By using the modal reduction method, the system equations of motion in physical coordinates are transformed into the linear modal coordinates and the sizes of the system matrices are reduced drastically. The reduced system modal equations can be handled easily with less computational efforts. The jBrequency-maximum deflection relations of simple harmonic, superharmonic and subharmonic responses are predicted by choosing suitable initial conditions. The procedure for the selection of the initial conditions is also presented. A laminated composite plate is studied in great detail. External loadings considered are harmonic excitations or combined harmonic and thermal loads. The steady state responses of the linear modal coordinates are presented in details at several frequencies. Their phase plots, power spectrums and time domain graphs are given and discussed. Introduction The increase use of advanced composites as high performance structural components necessitates accurate prediction methods which reflect their multilayered anisotropic behavior. Thin laminated composite plates subjected to severe harmonic lateral loadings are likely to encounter flexural oscillations having amplitudes of the order a plate thickness. For the prediction of forced vibration response, the multilayered anisotropic behavior, the complex boundary conditions, and the complex loading cases such as the present of the thermal loads make the problem even more difficult. Methods of analysis dealing with large deflections are thus becoming increasingly important. Whitney and Leissa [1] have formulated the basic governing equations for nonlinear vibrations of heterogeneous anisotropic plates in the sense of von Karman. Based on those equations, a number of classical continuum 1423 approaches exists for the analysis of nonlinear plate behavior. In general, the Galerkin’s method is used in the spatial domain, where the plate deflection is expressed in terms of one or more linear vibrational mode shapes; and various techniques in the temporal domain such as the direct numerical integration, harmonic balance, incremental harmonic balance, perturbation, and multiple scales methods, to cite a few, are employed. Excellent collections of classical continuum solutions and reviews on geometrically nonlinear analysis of laminated composite elastic plates are given by Chia [2,3] and Sathyamoothy [4]. The internal resonance of nonlinear systems has been thoroughly investigated using the multiple scales by Nayfeh and Mook [5]. Most recently, Wolfe et al. [6] have reviewed various analytical methods and have obtained experiment data on beams and plates excited sinusoidally or randomly. Most of the classical continuum solutions of composite plates have been limited to single-mode approximation. This is due to the difficulties in obtaining the general Duffing-type multiple-mode equations using the Galerkin’s approach especially for arbitrarily (unsymmetrically) laminated composite plates with complex boundary conditions. The finite element method has proven to be a powerful and versatile approach for structural problems of complex geometries, boundary conditions, and loadings. Reddy [7] has reviewed the application of finite element methods to linear and nonlinear anisotropic composite plate problems. In this paper, the nonlinear steady state periodic responses of thin rectangular arbitrarily laminated composite plates excited sinusoidally with or without the presence of thermal load are presented using the finite element time domain modal method. A rectangular composite plate is studied in detail. Formulation The finite element system equations of motion for large amplitude vibrations of a thin laminated composite plate can be expressed as [M]{w}+[C]{w}+([K]-[Knt]+[K 1(W)]+[K2(W)]){W} = {P(t)}+{PT} ( 1 ) where [M], [C], [K], [Knt] , {P(t)} and {Pt} are constant matrices and vectors and represent the system mass, damping, linear stiffness, thermal effort and 1424 loads, respectively; and [Kl] and [K2] are the first and second order nonlinear stiffness matrices and depend linearly and quadratically on the unknown structural nodal displacements {W}, respectively. The derivation of the element matrices and load vectors and their explicit expressions are referred to references [8,9]. The system equations of motion presented in eq. (1) are not suitable for direct numerical integration because: a) the nonlinear stiffness matrices [Kl] and [K2] are functions of the unknown nodal displacements, and (b) the number of degrees of freedom (DOF) of the system nodal displacements {W} is usually too large. Therefore, eq. (1) has to be transformed into the modal or generalized coordinates of much smaller DOF. Various reduction methods for nonhnear problems have been summarized in an excellent review article by Noor [10]. For nonlinear dynamic problems, the base vectors need updating using the modal methods presented in [10]. In the present formulation, the forced general Duffing-type modal equations will have constant nonlinear modal stiffness matrices, therefore updating of the base vectors is not needed. This is accomplished by a modal transformation and truncation {W} = ^q,(t){(|)}">=[<I>]{q} (2) r=l where the system mode shapes are the solution from the linear eigen-problem cOf [M]{(j)}^^^ =[K]{({)}^^\ The nonlinear stiffness matrices [Kl] and [K2] in eq. (1) can now be expressed as the sum of the products of modal coordinates and nonlinear modal stiffness matrices as [Kl] = ^qr[Kl«)W)] (3) r=l and [K2] = ^ ^q^qs [K2((j)<">(t)<^’)](4) r=l s=l The nonlinear modal stiffness matrices [Kl]^"^ and [KZ]^"®^ are assembled from the element nonlinear modal stiffness [kl]^''^ and [k2]^^^^ as 1425 (5) ([K1]">,[K2]'”>) = ^([kl]® ,[k2]<“>) all elements + bdy. conds. where the element nonlinear modal stiffness matrices are evaluated with known system linear mode W. Thus, the nonlinear modal stiffness matrices and are constant matrices. Equation (1) is thus transformed to the forced general Duffing-type modal equations as [M]{q}+[c]{q}+([K]+[Klq]+[K2qq]){q} = {F(t)} (6) where the modal mass, damping, and linear stiffness matrices are ([m], \c\ [k]) = [OJ'T ([M], [C], [K] - [Kot ])[<I.] (7) and the quadratic and cubic terms in modal coordinates and the modal force vector are [Klq]{q} = [<E.f 2^qr[Kl]« lr=l [.K2qq]{q} = [3.]^ ^^q,q,[K2] [<E>]{q} Vr=l s=l ( 8 ) (9) {F} = [cI>f({P(t)}+{PT}) (10) AH modal matrices in eq. (6) are constant matrices. With given initial conditions, the response of modal coordinates {q} can be determined from eq. (6) with any direct numerical integration scheme such as the Runge-Kutta or Newmark-P method. Therefore, no updating of the vibration modes is needed. The following is the description of the selection of the initial conditions for periodic motions. With the input of suitable initial conditions, three types of solutions, periodic or nearly simple harmonic, superharmonic and subharmonic solutions, can be obtained. The selection of each type solution is based on the solution of the 1- 1426 DOF Duffing equation obtained by using the modal reduction method described earlier. For example, the system equations of motion of a symmetric composite plate can be reduced to 1-DOF model (Note: the quadratic term is gone because the plate is symmetric) as Mjq,+Crqr+Krqr+K2,q^ =FrSin(C0t) (11) where Mr, Cr, Kr, K 2 r and Fr are scalar constants and represent the modal mass, damping, stiffness and force; co is the forcing frequency and the subscript “r” denotes the linear modal number. The solution of eq. (11) can be assumed as =AiCos(cot)+ A 3 cos( 3 cot) for the simple harmonic and superharmonic solutions, then two sets of Ai and A 3 can be obtained by the substituting of the assumed qr into eq.(n). One set is for the simple harmonic solution and the other set is for the superharmonic solution. Based on these solutions, the initial displacement of the r-th modal coordinate in eq. (6) is chosen as A 1 +A 3 . The initial velocities and all other initial displacements are zero. Similarly, it is assumed that qr =Aicos(cot)+ Ai/3Cos(cot/3) for the subharmonic solution. Then, all those initial conditions can be found by repeating the procedure just described. Results and Discussions A simply supported eight-layer symmetrically laminated (0/45/-45/90)s composite plate is studied in great details. The plate isofl5xl2x 0.048 in. (38.1 X 30.5 X 0.122 cm). The inplane boundary conditions are immovable, i.e. u=v=0 on all four edges. The graphite-epoxy material properties are : Ei=22.5 Msi (155 GPa), E 2 =1.17 (8.07), G 12 = 0.66 (4.55), V 12 = 0.22 and p = 0.1458 X 10'^ ib-sVin^ (1550 kg/m^). The C‘ conforming rectangular plate element is used in the finite element model and the plate is modeled with 12 x 12 (144 elements) mesh. The element has a total of 24 DOF (16 bending and 8 membrane). The lowest six natural frequencies (cOr, r=l,6) and their corresponding mode shapes are : coi = 55.46 Hz for (1,1) mode, Oh = 125.736 Hz for (2,1) mode, CO 3 = 151.951 Hz for (1,2) mode, CD 4 = 216.475 Hz for (2,2) mode, CO5 =250.585 Hz for (1,3) mode and cOe = 310.774 Hz for (3,1) mode. 1427 Two load cases considered are uniformly distributed harmonic excitation over the plate with and without the piresence of temperature. A constant modal damping factor, = C,/(2Mr 0)r), of 0.02 and a four-mode solution are used in the examples (Only the (1,1), (2,2), (1,3) and (3,1) modes are considered because the uniformly distributed excitation cannot induce any response of the (1,2) and (2,1) modes, see Table 7 of [11]). Harmonic Excitations A uniformly distributed pressure load of the form p(x,y,t) =po sincot is considered for the forced vibration problem. The force intensity is maintained at po = 0.00438 psi (30.2 Pa), however, the forcing frequency co is varying in a wide range from 0 to 4.5 times of the lowest linear natural frequency ©i. The results are shown in Figs. 1 to 4, where the designations for the total responses are indicated in Fig. 1, while the time-histories, phase plots and power spectra are given in Figs. 2 to 4. To make clear the behavior of the vibration response at particular frequency, each modal coordinate is depicted for understanding the simple harmonic, superharmonic and subharmonic response of the nonlinear system. Figures 2a-c correspond to the responses of the three forcing frequencies at 0.6, 2.4 and 3.8 times of ©i labeled as (1) to (3) in Fig. 1. It can be seen that the total response of the centre of the plate is dominated by the first mode (It should be noted that the centre of the plate has zero contribution from the (2,2) mode). The frequency responses of the four modal coordinates are composed of superharmonic frequency components of order 2, 3, 5 .. etc. , as well as the input driving force frequency. At the frequency of the point (3) in Figs. 2c and 2 d, the corresponding time histories of the second, third and fourth modal coordinates are unsymmetric as that of the first modal coordinate is symmetric. Hence, the plate is vibrating with a non-zero equilibrium position due to the unsymmetric responses of the second, third and fourth modes. In Figs. 3a and 3b, which correspond to the responses of two points (4) and (5) at © = 2.4 ©1 and 4.2 ©i in Fig. 1, the total response of the centre of the plate is almost pure simple harmonic at that particular frequency range. In Fig. 4a, which corresponds to the response of the point (6) at © = 3.8 ©i in Fig 1, the centre of the plate is mainly composed of subharmonic response of order 1/3. In the time histories of the four modal coordinates of Fig. 4b, it can be seen that the subharmonic component in the total response is contributed by 1428 the (1,1) modal coordinate, and the responses of the higher modal coordinates are pesudo harmonic. Combined Harmonic and Thermal Loads In addition to the uniform pressure po sincot, a steady state temperature change of 2.9Tcr is also applied to the composite plate ( where the buckling temperature =13.79 °F). The forcing frequency is taken as co = 1.45c0i and it is kept at that constant frequency of excitation, however, three pressure intensities at po = 6, 10 and 14 x 10*^ psi are considered. The responses are shown in Figs. 5a-c. The results are shown after the transient response being damped out, this is demonstrated by the quasi-steady state time histories in Figs. 5a and 5c. When the pressure load is small at 0.006 psi (41.3 Pa), the plate exhibits small oscillations about one of the thermally buckled positions (Wmax/h =1.0237) shown in Fig. 5a. With increase of the pressure loading, the amplitude of vibration increases. Fig. 5b shows the so-called snap-through or oil-canning phenomenon at po = 0.010 psi (68.9 Pa), the plate behavior is chaotic and has two potential weUs. With the further increase of the pressure loading, the plate exhibits large amplitude oscillations through the two buckled positions as shown in Fig. 5c at po = 0.014 psi (96.4 Pa). The plate motion is periodic at low and high pressure loads, however, the plate response composes of superharmonic frequency components of order 2 and 3 at the low pressure and of order 3 and 5 at the high pressure as shown in the PSD plots. The substance of the transition of the three distinct plate behaviors, from the small oscillations into the chaotic motion then into the large amplitude vibrations with the increase of forcing intensity, is shown in Fig. 6. In the low pressure range, the plate could also vibrate about the another equally possible bifurcation buckled position shown with dotted lines. Conclusion Based on the finite element method, a multimode time-domain formulation for nonlinear forced vibration of composite plates is presented. The main advantage of this method is that the system matrix equation derived from the FEM can be transformed into a set of general type Duffing equations with constant system matrices and much smaller DOF. The selections of initial 1429 conditions for the subharmonic, simple harmonic and superharmonic responses are presented. Through detailed descriptions, the frequency response characteristics, phase plots, time histories and the power spectrums have been illustrated for the three types of responses. The responses of a thermally buckled composite plate under harmonic excitation with fixed forcing frequency and various amplitudes are also obtained. Snap-through motion is observed at moderate pressure loads. References 1. J. M. Whitney and A. W. Leissa 1969 Journal of Applied Mechanics 36, 261-266. Analysis of heterogeneous anisotropic plates. 2. C. Y. Chia 1988 Applied Mechanics Review 41, 439-451. Geometrically nonlinear behavior of composite plate: A review. 3. C. Y. Chia 1980 Nonlinear Analysis of Plates, McGraw-Hill, New York. 4. M. Sathyamoorthy 1987 Applied Mechanics Review 40, 1553-1561. Nonlinear vibration analysis of plates: A review and survey of current developments. 5. A. H. Nayfeh and D. T. Mook 1979 Nonlinear Oscillations, John Wiley, New York. 6. H. F. Wolfe, C. A. Shroyer, D. L. Brown and L. W. Simmons 1995 Technical Report WL-TR-96-3057, Wright Laboratory, Wright Patterson AFB, Ohio. An experimental investigation of nonlinear behavior of beams and plates excited to high levels of dynamic response. 7. J. N. Reddy 1985 Shock and Vibration Digest 17, 3-8. A review of the literature on finite element modeling of laminated composite plates. 8. Y. Shi and C. Mei 1996 Proceedings of the 37th AIAA / ASME / ASCE / AHS / ASC Structures, Structural Dynamics, and Material Conference, Salt Lake City, UT, 1355-1362. Coexisting thermal postbuckling of composite plates with initial imperfections using finite element modal method. 9. C. K. Chiang, C. Mei and C. E. Gray, Jr. 1991 Journal of Vibration and Acoustics 113, 309-315. Finite element large-amplitude free and forced vibrations of rectangular thin composite plates. 10. A K. Noor 1981 Composites and Structures, 13, 31-44. Recent advances in reduction methods for nonlinear problems. 11. Y. Y. Lee, Y. Shi and C. Mei 1997 Proceedings of the 6th International Conference on Recent Advances in Structural Dynamics, University of Southampton, UK. A finite element time domain multi-mode method for large amplitude free vibration of composite plates. 1430 Centre Disp/Thlckness Centre Disp/Thickness Po=0.00438 Psi Damp. Ratio = 0.02 (3) ( 2 ) ( 1 ) ■ ■ (4) (6) (5) 0 1 2 oa/osl 3 4 5 Freq. Ratio Figure 1. Frequency response of the simply supported (0/45/-45/90)s rectangular plate 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Po (psi) Figure 6. Plate centre response vs force amplitude at co = 1.45ci>i and T = 2.9 Ter 1431 1432 1433 1434 1435 1436 GEOMETRICALLY NONLINEAR RESPONSE ANALYSIS OF LAMINATED COMPOSITE PLATES AND SHELLS C.W.S. Tot and B. Wangt Department of Mechanical Engineering University of Nebraska 255 Walter Scott Engineering Center Lincoln, Nebraska 68588-0656 U.S.A. E-mail; cwsto@unlinfo.unLedu Abstract The investigation reported in this presentation is concerned with the prediction of geometrically large nonlinear responses of laminated composite plate and shell structures under dynamic loads by employing the hybrid strain based flat triangular laminated composite shell finite elements. Large deformation of finite strain and finite rotation are emphasized. The finite element has eighteen degrees of freedom which encompass the important drilling degree of freedom at every node. It is hinged on the first order shear deformable lamination theory. Various typical laminated composite plate and shell structures under dynamic loads have been studied and representative ones are presented and discussed in this paper. Shear locking has not appeared and there is no zero energy mode detected in the problems studied. It is very accurate and efficient. Consequently, it is relatively much more attractive than other elements currently available in the literature for large scale nonlinear dynamic response analysis of laminated composite plate and shell structures. t Professor and corresponding author i Research Associate 1437 1. INTRODUCTION Many modem structures such as nuclear reactor containment installations, naval and aerospace structures, and their components, must be designed to withstand a variety of intensive dynamic disturbances. Because of their many attractive features over isotropic materials more and more stmctures or components in the aforementioned systems are made of laminated composite materials. The investigation reported in this paper is therefore concerned with the prediction of geometrically large nonlinear responses of laminated composite plate and shell structures, of complicated geometries, under transient excitations. With complicated geometries analytical solution is impossible and therefore a versatile numerical method, the finite element method has been employed. A hybrid strain based flat triangular laminated composite shell finite element has been developed by the authors [1,2] for the nonlinear analysis of plate and shell structures under static loadings. The present investigation is an extension of [1,2] to cases with the aforementioned dynamic forces. Among various attractive features of the derived element stiffness and consistent element mass matrices five are worthy of listing here for completeness. These are: (a) their ability to deal with large nonlinear elastic response of finite strain and finite rotation, (b) the fact that they are in explicit expressions and therefore no numerical integration is necessary, (c) the obtained results of a relatively comprehensive tests [2, 3] show that the element is free from shear locking, (d) the element gives correctly six rigid body modes, and (e) the finite element has three nodes and eighteen degrees of freedom (dof) which encompass the important drilling degree of freedom (ddof) at every node. It is based on the first order shear deformable lamination theory. It is a generalization of the low-order flat triangular shell element for isotropic materials developed earlier by Liu and To [4]. It is noted that one of the earlier work that employed triangular shell element is due to Noor and Mathers [5]. In the latter a mixed type triangular element was proposed. The element has six nodes, and 78 dof. It was based on the shallow shell theory and was shear deformable. Recently, Lin et al [6] developed a finite element procedure to analyze composite bridges. The finite element procedure was based on small elasto-plastic strains and updated Lagrangian formulation. The element used was flat and constructed by the superposition of a discrete Kirchhoff bending element and a linear strain triangular membrane element. It has six nodes. There are three translational and three rotational dof at its comer nodes and three translational dof at mid¬ side nodes. In 1994 a flat triangular shell element was presented for static nonlinear analysis by Madenci and Barut [7]. It is based on the so-called free formulation concept for analyzing geometrically nonlinear thin composite shells. A corotation form of the updated Lagrangian formulation is utilized. The theoretical basis was on the geometrically nonlinear Kirchhoff plate theory without considering the effects of transverse shear deformation. The element 1438 is of displacement type. It has three nodes and six dof for each node. While such formulation has some advantageous features computationally the element is relatively less efficient because (a) the linear element stiffness matrix consists of a basic and a higher-order stiffness matrices in the sense of Bergan and Nygard [8] for isotropic materials, and (b) the important effects of transverse shear deformation in the plate component of of this element has been disregarded. A more recent contribution on triangular elements is made by Zhu [9]. The natural approach is used to construct a curved triangular shell element for static analysis of geometrically nonlinear sandwich and composite shell structures. The element has six nodes. There are six dof at each corner node and three dof at each mid-side node. Updated Lagrangian description was adopted in the procedure. In the element formulation the transverse shear deformation was considered by assuming constant transverse shear stress distribution. In the next section the formulation of element stiffness matrices is outlined. Section 3 deals with the derivation of element matrices. Section 4 is concerned with the application of the derived elements to three example problems of plate and shell structures. The concluding remarks are included in Section 5. 2. FORMULATION OF ELEMENT STIFFNESS MATRICES Finite element formulation for the derivation of a family of simple three-node, six dof per node, hybrid strain based laminated composite triangular shell finite elements for large scale geometrically nonlinear analysis is briefly outlined in this section. Large deflection of finite strains and finite rotations are included. The first order shear deformation theory and the degenerated three dimensional solid concept are adopted. In particular, element matrices for one member of the family are derived explicitly with the symbolic computer algebra package MACSYMA. To minimize the algebraic manipulation involved in the derivation, updated Lagrangian description is employed in the incremental formulation of the finite element procedure. In essence, the present formulation is an extension of the work by Liu and To [4] for isotropic materials to multi-layer laminated composite shells. Therefore, in the development the present approach follows closely that of the last reference. 2.1 Incremental variational principle The Hellinger-Reissner functional tChr can be written as where, e' is the independently assumed strain field; e“ is the strain due to displacement; C is the material stiffness matrix or elasticity matrix; W is the work done by external forces, and the superscripts e and u indicate that the quantities are from independently assumed strain field and displacement field, respectively. For geometrically nonlinear analysis with incremental formulation and updated Lagrangian description, the static and kinematic variables in current equilibrium configuration at time t are assumed to be known quantities and the objective is to determine their values in the unknown subsequent equilibrium configuration at time t+At. For a time increment At, that is from time t to (t+At), one has - ATTjjj^CAUjAe®) - 7ifjjj(t+At) - ttHR® , or, with reference to equation (1), Atihr " / [(e')TC(Ae“) - i(Ae‘)fC(Ae') - where, Au is the vector of incremental displacement; Ae* is the vector of independently assumed incremental updated Green strains; Ae“ is the vector of incremental updated Green ’geometric’ strains or incremental Washizu strains; AW is the work-equivalent term corresponding to prescribed body forces and surface tractions in configuration C Equation (3) is the incremental form of Hellinger-Reissner variational principle. For updated Lagrangian description, the integral is evaluated at the current configuration C In the equation, the term J(Ae')^C(e^ - e“) dV Ve ■f (Ae'')'^C(Ae“) (3) (Ae')'^C(e‘^ - e'^)] dV - AW . 1440 is the so-called compatibility-mismatch. Numerical results of Saleeb et al [10] showed that though totally discarding the term resulted in convergence difficulties, while including the term in only the first iteration of every load step yielded essentially the same results as those having the term under all circumstances. However, Liu and To [4] reported no difficulties for convergence when the term was ignored. In the current study, this term is also disregarded. Then equation (3) can be recasted as ^^hr“/ [ + (Ae®)^C(Ae“) Vc (4) -■i(Ae')’'C(Ae')]dV-AW, 2 where = (e^)^C is the Cauchy (true) stress vector at the current configuration C ^ In this equation, the incremental Washizu strain Ae“ can be expressed in two parts Ae^ Aei" ^ ATii“ (5a) and they are related to the incremental displacement by |(Auy AUjj) . Arii” -AUyAU,^ (5b,c) where the Einstein summation convention for indices has been adopted and the differentiation is with respect to reference co-ordinates at the current configuration C Substituting equation (5a) into (4) yields A’'™ = f [- -(Ae')’'C(Ae') + (Ae')'^C(A£") + a'^Ae” V. 2 (6) + o^Aii" + (Ae')fCATi“ ] dV - AW , where a is the Cauchy stress vector. Discarding the higher order term, ) ^Aq ^ results in AjIhr “ / [- -(Ae')'^C(Ae') + (Ae')TC(AE“) + a^Ae'^ + a^Aq^] dV - AW . (7) 1441 2.2 Hybrid Strain Formulation Element stiffness matrices for a hybrid strain based finite element can be derived directly from equation (7). Generally the independently assumed strain field and displacement field can be written as Ae' = P Att , Au = <t) Aq where P is the strain distribution matrix, (|) is the displacement shape function matrix, Aa is the vector of incremental strain parameters and Aq is the incremental nodal displacement. Substituting equations (8a, b) into (7), and defining H - J P^CP dV^ , G, = / P'^CBl dV^ , V - / dVe . F. " / BlO dV. , V V, one can show that AT:jnj(Aq,Aa) = ’ ^Aa'^HAa + Aa'^G^Aq + F^Aq + ■iAq^'k^LAq - F'^Aq ] ( 10 ) where F is the external nodal force vector in the neighbour configuration associated with the AW term in equation (7); and B^l are the linear and nonlinear strain-displacement matrices, while Cq is the matrix containing the Cauchy stress components at the current configuration. Finally, one can show that (k[ . V) Aq - F{t. At) - F, , k[ = g7h-‘G. 0 where the expression in equation (1 lb) is the element "linear" stiffness matrix. The term k^L defined in equation (9) is the "nonlinear" or initial stress stiffness matrix and Fi is the pseudo-force vector. The right hand side of equation (11a) is the equilibrium imbalance. 1442 3. ELEMENT MATRICES AND THEIR UPDATING The derivation of nonlinear element stiffness matrices, constitutive equations, mass matrices of the element shown in Figure 1 are outlined here. In addition, updating of configuration and stresses at every time step is considered here for completeness. 3.1 Nonlinear Element Stiffness Matrices For the assumed displacement field, an arbitrary point within an element is governed by r*' 3 t h t Si 0 3 ( 12 ) The incremental displacements of an arbitrary point within the element are Au' 1 1 O Au/| 3 < II AVi 1 > + (AVi') . i-l Aw' , 1 Aw/ 1*1 Employing quadratic polynomials for the translational dof and including ddof lead to Au' ^ 3 AUi' 3 Ai(ii) Ai(i2) A0' Av' Aw' = E?i^ i-l AVi' Awj' ^ c'E^i i-l Ai(21) Ai(22) Ai(31) Ai(32) ^ 1-if 0 0 Pi AO^i 0 0 5i A Aei • . "Pi -5i 0 Ae' (14) 1443 In the foregoing, U = Ui^l + U2^2‘^U3^3 + Piejl + p20t2-"P3®G W = WiCl+W2^2-"W3^3-Pie,l-P2e ^- 536,3 " 5 i®s 1 “^ 2 ®s 2 “^ 3 ®s 3 » 0r = 0rl^l'*'®i2^2-*'®r3^3 ®s*®sl^l ■*'®s2^2'^®s3^3 ®tl^l '*’^ 12^2 ®e^3 (15a) (15b) where Pi =■ (^1^3 "^12^2)^! ’ 5i “ (^31^3 “^12^2)^! » P2 * (^12^1 ”^3^3)^2 » ^2 * (^12^1 “^23^3)^2 ’ P3 “ (^3^2 ~ ^31 ^1)^3 » ^3 ” (^23^2 1)^3 • The remaining symbols have been defined by Liu and To [4] and are not repeated here for brevity. For the assumed strain field, the strain vector in equation (8a) may be written as Ae*^ Ae; 4 Ay' (16) where A^'m = PmA«„ . Ax' - PfcAa, Ay' - PsA«s (17a) 1444 with Aa^-lAttj Aa 2 Actj)’'’ , (17b) Aa^-{Aa4Aa5Aa^}'^,Aa^-{Aa7Aa8Aa,}’', and 1 0 o' 1 0 o' B II 0 1 0 11 pf 0 1 0 o, 0 1. 0 0 1 - 83 ( 1 - 2 ^ 2 ) 83 ( 1 - 2 ^ 1 ) 0 -13(1-2^2) (r3-r2)(l-2^i) 12(1-2^3) where the subscripts m, b and s denote the membrane, bending and transverse shear components of P in equation (8). By defining = / PjA'P^ da , = f P3"C;P„ da , a a - / Pjc^P. da , - / Ps^E'P, da , a a = / PjB'P^ da , = / P,"CJP, da , (18) a a H„b - / PmB'Pb da , P,^CX da . a a Hbb = / Pb'D'Pb da 1445 where a is the area of the triangular shell element and A' - E (C.)k(h,-h,.,). B' - i ^ (C,)k(h^h,l,), k-1 ^ k-1 " 'T S (^a)k(^k“'^k-l) ’ (^b)k(^k“^k-l) ’ d k-1 k-1 Ca - i: (c j,aik“Vi) > Cb - -i i: (c k-1 k-1 (19) in which the integer n is the number of laminae in the laminated composite structure. Then the matrix H in equation (9) becomes H = Hta Hu H* 9x9 ( 20 ) Similarly by defining / pJa'B^ da . Gsm - / Ps"cX <ia a a Gms ■ / da , = / Ps^E'B, da , a a Gbm = / P,^B'B„ da , Gi, ■= / pXb, da . a a Gmb “ / pJb'B, da , G., - / PsXX ^ . a a G,, = / pJd'B, da 1446 one has the matrix + G„b + G^ B X) o + Gbb + Gbs Gs. + G.b + 9x18 ( 22 ) Therefore, with the ddof considered the element stiffness matrix can be shown to be k - k[ + k^ + k^L (23) where the linear element stiffness matrix k^’ and the "nonlinear" or initial stress stiffness matrix k^L ^re defined by in equations (9) and (11), while the stiffness matrix associated with the ddof k^d is defined as kda = i:(G.),(h,A-.)/B7B,da, (24) k-1 in which Bd = [Bdi Bj2 B^Ji^ig and Bdi -Ui.r 0 0 0 with i = 1,2,3. The "nonlinear" or initial stress stiffness matrix k^L can be obtained if the nonlinear strain-displacement matrix B^l and the matrix Gc which contains the Cauchy stress components at the current configuration are available. The matrix B^l is defined by equation (45) of Liu and To [4]. The matrix Oq is constructed from the Cauchy stress vector c and defined as ^11^ °3ll3 ^22^3 ” 23 '^ *^23^ O 3 (25) 1447 with I 3 being the 3x3 identity matrix and O 3 a 3x3 null matrix. The transverse stress components of a are considered constant over the thickness, and all components of a are calculated and updated for each time step at the centroid of each element. 3.2 Constitutive Equations For finite strain problems in the elastic range, the reduced stiffness matrix is a function of stresses. To incorporate finite strains in the analysis, several approaches can be applied. The following adopted from reference [4] is to add the linear elastic matrix a correction matrix which is a function of Cauchy stress. To begin with, the correction terms in tensor form becomes Ciid " - * OjiSfl + OnSik + "jiSik ) where 8 ^^ is the Kronecker delta. Note that this equation comes as a result of transforming the Jaumann stress rate to the incremental second Piola-Kirchhoff stress. If the stress and strain vectors are O = { Ojj O 22 O 33 0^2 ^23 Ojj } , e = { ®22 ®33 ®12 ®23 ®31 the matrix form of equation (26) is 0 0 2012 0 2013 0 4<J22 0 20i2 20,3 0 1 0 0 4033 0 2023 2013 2 20i2 2a,2 0 ^ll'^^22 Oi 3 ^23 0 2°23 2023 °13 ^22"^ ^33 ^12 2oj3 0 20j3 ^^23 ^12 Oii + O In present investigation the so-called degenerated concept is adopted and therefore the elastic modulus in the normal direction to the plane of the shell structures is considered zero. Consequently the stress and strain in the transversal direction are ignored. In the linear analysis the constitutive relations for a lamina have been defined as o = Qe (29) 1448 where °n ° zx ’ e - { e, By e„ )■" Qn Qi 2 Qi 6 0 0 Qi 2 Q22 Q26 0 0 Qi 6 Q26 0 0 0 0 0 Q44 0 0 0 ^5 Q55 The corresponding matrix from equation (28) is 0 2<’.y 0 2°. 0 40 y 2<^.y 2o yz 0 2% °yy 0 2^. ® zx °xy 0 ^xy The material stiffness matrix for a lamina thus becomes C = Q + C (30) (31) (32) (33) where t^aJ3x2 t^aj2x3 J2x2. in which C„ C^, and = ^ba are given in equation (19). With the consideration of large deformation and finite strain, the constitutive equations for a multilayered structure or laminate can be written as 1449 or simply (34) ' N ' ■ A' B' Ca ^ M D' Cb - X Qs Ub e' 8x8 . y . ^ ®N where N, M and Qj are the vectors of stress resultants corresponding to membrane, bending and transverse shear, respectively. The matrices A’, B’, D’, E’, and Cg have been defined in equation (19). 3.3 Element Mass Matrices and Updating of Configurations and Stresses In the present study, with the updated Lagrangian description, the consistent mass matrix is formulated in the current configuration C'. The mass matrix is then updated at each time step. The assumptions are that the angular velocities and accelerations are small enough to be discarded. By following the procedures of Liu and To [4] the consistent element mass matrix can be obtained as in which mt„ and m„t are translational and rotational components of the consistent element mass matrix, respectively. Matrix m^ is the part associated with the ddof. When it is used for the incremental formulation with updated Lagrangian description, updating relevant quantities at each incremental step are required before evaluating the mass matrix. All these mass matrices are obtained explicitly with the symbolic computer algebra package MACSYMA. For each incremental step, the configuration and stresses have to be updated. Details of the steps can be found in the reference by Liu and To [4] and therefore are not included here. However, it may be appropriate to point out that the linear consistent element matrix for multi-layer composites has been employed by the authors [11] for vibration analysis of plates and shells. 4. EXAMPLES OF LAMINATED COMPOSITE PLATE AND SHELL There are two main objectives in this section. First, accuracy of results obtained by the presently derived element matrices is studied. Second, the validity and conceptual adequacy of the formulation and assumptions made in the derivation of element matrices are assessed. For brevity, one multi-layer plate, one multi-layer shell structure, and a cantilever panel with free end step moment are included here. More example problems can be found in To and Wang [3], and Wang and To [12]. 4.1 Multi-Layer Plate Under Uniformly Distributed Step Disturbance The square plate considered has two layers. Its geometrical dimensions are: side length a = 2.438 m and total thickness h = 0.00635 m. Each layer of the laminate has equal thickness. The plate stacking scheme^is cross-ply (0/90). The layer material properties are: ^ = 6.8974 x N/m^, Ej = 25 Gi 2 = G ,3 = 0.5 E 2 , G 23 = 0.2 E 2 , V 12 = 0.25 and density p = 2498.61 kg/m . It is supported by hinges at its four edges. At these edges U or V (note henceforth upper case of deformation variable refers to global co-ordinate) parallel to the edges are not constrained. These boundary conditions are denoted as BCl in reference [13]. For the purpose of direct comparison with the results reported in the latter reference, one quarter of the plate is modeled by a 4 X 4 D mesh (see Figure 1 for the definition of D mesh). Thus, the boundary conditions applied are: V = 0 ^ = 0.0 at AB, U = W = 0,, = 0.0 at BC, V = W = 0y = 0.0 at CD and U = 0y = 0.0 at AD. In addition, all 0, are constrained. After application of the boundary conditions there are 158 unknowns in this case. The uniformly distributed transversal step disturbance with intensity po = 490.5 N/m^ is applied to the plate. In the analysis, the option of inclusion of directors [3, 12] and small strain are selected. The time step size is At = 0.001 seconds. The responses at the centroid obtained by using the HLCTS element are plotted in Figure 2. They are compared with those reported by Reddy [13] in which results were obtained with a nine-node rectangular isoparametric element. In the latter transverse shear was considered. Excellent agreement can be observed. Before leaving this subsection it may be appropriate to mention that the nonlinear element stiffness matrix presented in reference [13] is nonsymmetric while the one derived in the present investigation is symmetric. In fact, when the system is conservative the nonlinear element stiffness matrix can be shown to be symmetric. 4.2 Spherical Shell Segment Under A Uniformly Distributed Step Disturbance The geometry of the spherical shell is shown in Figure 3 in which the shell is simply supported. The geometrical properties are: radius R = 10.0 m, the side length of the projected plane b = 0.9996 m and the total thickness h = 0.01 m. The spherical shell is considered having two equal thickness layers and they have the (- 45/45) lamination scheme. The pertinent material properties are: Ej — 2.5x10^' N/m^ E 2 = 1.0x10^° N/m^, Gi 2 = G 13 = 0.5x10 ^ N/m^, G 23 = 0.2x10*° N/m^, Poisson’s ratio V 12 = 0.25 and density p = 1.0x10 kg/m^. For comparison to results available in the literature one quarter of the 1451 shell is modeled by the proposed hybrid strain based shell element (identified as HLCTS for brevity and convenience) with 4 x 4 D mesh. The boundary conditions applied to the finite element model are: V = = 0.0 at line AB,V = W = ©, = 0.0atBC,U = W = ©y = 0.0 at DC and U = ©y = ©, = 0.0 at AD. The number of equations to be solved after the application of the boundary conditions is 189. A distributed step pressure is applied to its outer surface (pointing toward the outer surface). It has an intensity p = 2000.0 N/ml The time step used were 0.03 s, 0.01 s and 0.005 s. As there was no significant difference and for efficient reason throughout the computation the time step of 0.03 s was adopted. The nonlinear transient response at the apex (central point A of the shell) is obtained and plotted in Figure 4. The problem has been solved by Wu et al. [14] who applied a curved high-order quadrilateral shell element. The latter has 48 dof and was developed based on the classical lamination theory. It is observed that there is a discrepancy of about 8 %, with respect to the HLCTS element results, for the amplitudes between the two set of results. However, they have the same vibration period. It is believed that the present results are more accurate as the element used in the present investigation is shear deformable. 4.3 Cantilever Panel With Free End Step Moment To demonstrate the use of the proposed shell element for structures undergoing large rotation and large deformation a four layer cross-ply cantilever panel is considered here. More computed results for this case can be found in references [3] and [12]. It is symmetrically laminated with the stacking scheme (0/90/90/0). Its geometrical properties are: L = 1.2 m, b = 0.1 m and h = 0.01 m. The material used for this cantilever is the high modulus graphite/epoxy composite. Its properties are: Ej = 2.0685x10" N/m^, E 2 = 5.1713x10'' N/m^ G,-, = 3.1028x10^ N/m^ G,, = G 23 = 2.5856x10^ N/m^ p = 1605 kg/m^ and Poisson’s ratio Vj 2 = 0.25. A step moment M about an axis parallel to the width of the panel is applied to the free end. The amplitude of this moment is = 1000.00 N-m. The panel is discretized by a 12 x 1 A mesh. At the fixed end, all dof are constrained. The finite element model has 144 unknowns. The time step At = 0.001 s is employed in the trapezoidal rule direct integration. The nonlinear transient responses at the end of the cantilever are solved by selecting the options of director included, small strain and constant thickness in the digital computer program developed. The computed end deflections are plotted in Figure 5. As noted in reference [3,12], the inclusion of directors in the formulation [15] is crucial as the directors are important parameters that constitute the so-called "exact geometry" for large rotation problems. 1452 5. CONCLUDING REMARKS The hybrid strain based laminated composite flat triangular shell (HLCTS) element for the static analysis of geometrically nonlinear laminated composite plates and shells has been further developed and employed to solve various dynamic problems. A relatively comprehensive study for various plate and shell structures idealized by this element has been performed and three representative examples are included to demonstrate its accuracy, efficiency and conceptual adequacy. It is concluded that the HLCTS element is attractive for large scale finite element analysis and modelling of shell structures undergoing geometrically large deformation of finite strain and finite rotations. ACKNOWLEDGMENT The first author gratefully acknowledges the financial support in the form of a research grant from the Natural Sciences and Engineering Research Council of Canada. 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Noor, A.K. and Mathers, M.D., Nonlinear finite element analysis of laminated composite shells. In Computational Methods in Nonlinear Mechanics (Ed. by J.T. Oden, E.B. Becker, R.R. Craig, R.S. Dunham, C.P. Johnson and W.L. Oberkampf). Proc. Int. Conf on Comput. Methods in Nonlinear Mechanics, Austin, TX, 1974. 6. Lin, J.J., Fafard, M., Beaulieu, D. and Massicotte, B., Nonlinear analysis of composite bridges by the finite element method. Comput. Struct., 1991,40. 1151-1167. 1453 7. Madenci, E. and Bamt, A., A Free-formulation-based flat shell element for nonlinear analysis of thin composite structures. Int. J. Numer. Meth. Engng., 1994, 37, 3825-3842. 8. Bergan, P.G., and Nygard, M.K., Nonlinear shell analysis using free formulation finite elements. In Finite Element Methods for Nonlinear Problems, (edited by Bergan, P.G., Bathe, K.J., and Wunderlich, W.) Springer- Verlag, 1986. 9. Zhu, J., Application of natural approach to nonlinear analysis of sandwich and composite plates and shells. Comput. Meth. Appl Mech. Engng., 1995, 120, 355-388. 10. Saleeb, A.F., Chang, T.Y., Graf, W. and Yingyeunyong, S., A hybrid/mixed model for nonlinear shell analysis and its applications to large- rotation problems, Int. J. Num.. Meth. Engng., 1990 29, 407-446. 11. To, C.W.S. and Wang, B., Hybrid strain-based three-node flat triangular laminated composite shell elements for vibration analysis. J. Sound and Vibration (submitted), 1996. 12. To, C.W.S. and Wang, B., Transient response analysis of geometrically nonlinear laminated composite shell structures. Proc. of Design Eng. Conf. and Computers in Eng. Conf. (edited by McCarthy, J.M.), August 18-22, 1996, Irvine, California, 96-DETC/CIE-1623. 13. Reddy, J.N., Geometrically nonlinear transient analysis of laminated composite plates. A.I.A.A. J., 1987, 21, 621-629. 14. Wu, C.Y., Yang, T.Y. and Saigal, S., Free and forced nonlinear dynamics of composite shell structures. J. Comp. Mat, 1987, 21, 898-909. 15. To, C.W.S. and Wang, B., Hybrid strain based geometrically nonlinear laminated composite triangular shell elements, Part I: Theory and element matrices. Computers and Structures (submitted), 1996. A mesh D mesh Figure 1 Flat triangular laminated composite shell element 1454 Central deflection lY Figure 2 Response of a cross-ply plate Figure 3 Spherical shell segment under a uniformly distributed load 1455 Central deflection -W Time b Figure 4 Apex response with quarter shell considered - 0,2 0.0 0.2 0.4 0.6 0.0 1.0 1.2 1.4 X (m) Figure 5 Evolution of cantilever panel with free end step moment 1456 ANALYTICAL METHODS III THE FREE, IN-PLANE VIBRATION OF CIRCULAR RINGS WITH SMALL THICKNESS VARIATIONS R S Hwang, C H J Fox and S McWilliam Department of Mechanical Engineering, University of Nottingham, University Park, Nottingham NG7 2RD, England Abstract Geometric imperfections which cause thickness variations will always exist in nominally circular rings and cylinders due to limitations in manufacturing processes. The effects of circumferential thickness variations on the natural frequencies of in-plane vibration are studied. The circumferential variations in the inner and outer surfaces are describedj in a very general wayj by means of Fourier series. Novozhilov thin-shell theory is used in conjunction with the Rayleigh-Ritz method to obtain the natural frequencies. Results are presented which show the effects of single-harmonic variations in the inner and outer surface profiles, taking account of the profile amplitude of, and the spatial phasing between, the inner and outer profiles. The frequency factors calculated from the numerical method are in good agreement with those obtained from the Finite Element method. 1. Introduction The free vibrations of circular rings or shells had been studied by many authors for over a century. The early theoretical works are summarised by Love [I]. Most of these works are restricted to perfect rings or shells. However, in practice, geometric imperfections (thickness variations and departure from true circularity) are produced in the manufacturing process. These affect the natural frequencies and mode shapes. It is weE known that in any truly axisymmetric structure the vibration modes occur in degenerate pairs which have equal natural frequencies and mode shapes which are spatially orthogonal but of indeterminate circumferential location. The main effects of thickness variations are to split the previous equal natural frequencies and remove the positional indeterminacy [2]. Although these effects are often practically unimportant, there are some applications (especially inertial sensors based on vibration rings or cylinders [3]) where the small frequency splits and fixing of the modal positions is of primary practical significance. There is therefore a requirement to be able to predict in detail the effects on vibrational behaviour due to small departures from perfect circularity of the kind produced by manufacturing tolerances. 1457 The vibration of imperfect bells and rings were studied m the general way using group theory [4,5]. In reference [5] the selection rules for frequency splitting of thin circular rings were presented qualitatively. In reference[6] the frequency splitting behaviour of a thin circular ring was investigated both experimentally and analytically by first order perturbation theory. In reference [7], the classical frequency equations, which are generally used to predict the natural frequencies of a thin circular ring, were modified to describe an eccentric ring by using the perturbation method. In reference [8], Fourier series functions were used to represent the circumferential thickness variations of an eccentric cylinder. Love thin-sheU theory, which is only strictly suitable for a perfect ring or cylinder, was applied to investigate the free vibration of non- circular shells. In this paper, the free in-plane vibrations of thin rings of rectangular cross section with circumferential variations in thickness are studied. The circumferential variations in the inner and outer surfaces are described, in a very general way, by means of Fourier series. Novozhilov thin-shell theory [9], in conjunction with the well-known Rayleigh-Ritz method, are applied to analyse the vibration characteristics for in-plane flexural vibration of the ring which is considered as a special case of a thin shell [2,6]. The numerical method is used to investigate the effect of single-harmonic circumferential variations in the inner and outer surface profiles. The effects of harmonic number, amplitude and spatial phasing between the inner and outer profiles are investigated. Some important trends and patterns of effects of profile variations on the splitting of the natural frequencies are observed. The results obtained by using the numerical method developed in the current investigation are validated by comparison with Finite Element predictions. 2. Method of Analysis 2.1 Geometry Consider a thin ring of mean radius having a rectangular cross- section of mean thickness h (« r«) and axial length L (« Ta). The inner and outer surface vaiy along the global circumferential direction (Figure 1). rp denotes the distance from the centre of the mean radius of the ring to the point F on the middle surface. Two coordinate systems are used in the formulation of the equation of deformation. 1458 * Global polar coordinates (a', p', These are dir^ted along the global axial, circumferential and radial directions. The initial geometry of the undeformed, imperfect ring is defined using this coordinate system. * Local curvilinear coordinates (a, p, These are directed along the local axial, tangential and normal directions relative to the true middle surface and coincident with the principal coordinates of the middle surface. This local coordinate system is required for implementation of Novozhilov shell theory which specifies displacements in the local tangential and normal directions. is the angle between the global and local coordinate systems at the point P of the middle surface. All the displacements, thicknesses, and radii in this paper are expressed dimensionlessly by dividing by lo , where k is the representative length and is defmed as the mean radius of the ring. Figure 1. A thin ring having circumferentially arbitrary surfaces The shape of the middle surface of the ring is determined by the inner and outer surfaces which can be expressed by Fourier series as follows: f*(?')=U+'^f*cos(ip) + '^f; sin(jp) (1) i=I j=I /TP’)=/«'+ X/r«osc«P'-)+X/7««o‘P') (2) i=/ 1459 where / and / "(P') denote respectively the outer and inner surface functions with respect to the global circumferential coordinate P and fo ,// j //, fo', fi' and// are the Fourier coefficients which are defined in the usual way [10]. The middle surface of a shell or ring is defined as the locus of the points which lie at equal distances, * and from the outer and inner surfaces along the direction normal to the mid surface (see Figure 2). .. ^ Figure 2. The bounding surfaces and the middle surface For given inner and outer surfaces, / (PpO and/'^(p/»0 defined in the global coordinate system, the true middle surface can be determined using an iterative numerical procedure which is fully described in [11]. Once the point P on the true middle surface has been determined, the corresponding , Rp, Tp and Pp can be calculated- These will be used in the step-by step integrations which determine the strain energy and kinetic energy of the ring. 2.2 Equations of Motion The strain energy for a thin ring whose length is much smaller than the mean radius takes the form [2,6]: S = «(!->■ l/R)d^d^ (3) 1460 Based on Novozhilov thin-shell theory, the normal strain epp in equation (3) is given as " 7 t / T> ^ (4) 1 + ^/ R Ep= Vj^/R + w/R (5) Kp= - 1/R ( IV, p/j? +v/R), p (6) where Ep, Kp characterise the deformation of the middle surface of the thin ring and subscript “ , p “ denotes partial derivatives with respect to (5. Epis the strain tangential to the middle surface and Kp is the change of curvature, v, w are the nondimensional local displacement components of the point P on the middle surface along the tangential and normal directions respectively. Substituting equations (4)-(6) into equation (3), then integrating with respect to the thickness from hi~ to neglecting the 4th and higher powers of /rF^ and ^ f and noting that F(p) [- hi~^]d?t = 0 where Ffp) is an arbitrary function of p, the strain energy of a thin ring can be derived in terms of the local displacements v as follows: S= {[( V, p/ + 2 h'v, p + w^]( 1 /R)[hi* -hi~ ] 2 rp; jK + [ 2WW, p + 2w, p IV, pp - 2v w - 2viv, pp ] -h 3K + [(w,^)^ + v‘.2vw,f]^ ((l/R),f)^[kC^ -hr^]}d^ ( 7 ) jK Similarly, the kinetic energy of a thin ring, based on Novozhilov thin- shell shell theory, can be expressed as follows: T= {[(y,,)\(y^,,f]R[hC + 2 '•Pi [3(v,t f- 4v,tW,,p + ( IV,/p f f (8) in which p is the density of the ring, and the subscript ‘,/ * denotes the partial derivatives with respect to time. 1461 For free vibration the tangential displacement v and the normal displacement w 'which satisfy the boundary condition can be assumed to take the following forms reg)ectively: V = sinnp-v^ cosnp)^^^^ (9) n w = cosnp + w^ sinnp)^^^^ (10) n where v„ and are the undetermined amplitude coefficients of the tangential and normal displacements of the middle surface respectively. The superscripts "s" and "c” refer to the fact that these coefficients are multiplied by sine and cosine terms respectively. Substituting equations (9) and (10) into equations (7) and (8), then applying the Rayleigh-Ritz procedure, the general frequency equation of the free ■vibrations for a thin ring is obtained and can be expressed in the following general matrix form: -A 'M" Ui~ ~0~ 0 where jfi: and M represent stiffiiess and mass matrices of size 2(N+1), and q denotes a vector of generalised coordinates v„, w„ etc. The matrix elements in equation (11) are given in [11]. Since in the general case the cross-section of the ring 'wdl not be symmetric with respect to P = 0, the classification of the modes as being “symmetric” and “antisymmetric” is meaningless. In the special case of a perfect circular ring, the off-diagonal terms [K^J, /M"/ and /Af7 appearing in equation (11) are null matrices, then equation (11) can be uncoupled into two equations: one is for the symmetric modes and the other is for the anti-symmetric modes -with respect to P=0. The frequency factors of the ring. An , are the eigenvalues of equation (11) and are defined by A 0)1 d P E ( 12 ) where a)„ is the natural frequency of the nth radial mode. The frequency factor An is proportional to the square of natural frequency . For a given value of n equation (11) will yield a pair of values of An. These will be equal in the case of a perfect ring but will be slightly different in 1462 the case of an imperfect ring, giving rise to a higher frequency mode and a lower frequency mode for each value of It should be noted that the matrix elements in equation (11) are expressed as integrals of the functions , Rp, yp and ^p with respect to the tangential coordinate These functions are expressed in terms of the local coordinates. Hence it is necessary to make a transformation to express these functions and integrals in terms of the global coordinates, so that the integrals can be evaluated over the global circumferential coordinate p* from 0 to 27t. 3. Results and Discussion By using different combinations of trigonometric functions in equations (1) and (2), it is in principle possible to model any closed thin ring. For the purpose of illustration we will consider a nominally circular ring with a single harmonic variation in the inner and outer surfaces, given by (P') ~ /" OT = + V (JP’ - <!>) where h/ and hf are the ampHtudes of the unperfections of the outer surface and the inner surface measured from the mean outer radius ra and the mean inner radius r/ respectively, is the spatial phase angle between the trigonometric functions of the inner and outer surfaces at p* =0 , and /, j are the harmonic numbers of the surface variations. Figure 3 illustrates i =] = 3 for three values of ^ . (^ = 0 <j> = 7t/2 ^ % Figure 3. Different spatial phase angles ^ for i = j = 3 1463 Results for the combinations of the geometric imperfections of i -J - 2, 3, 4, 5, 6, hf* - hf = O.lhy OMlh , and <J» = 0,7c/4, nil, SjtM, tz are presented here. The ring dimensions and material properties are as follows: = 40.75mm, r/ = 37.83mm, L = 2 ram = 206.7x10’ NW, p = 7850 kg/ml Note that hf = O.lh corresponds to a departure from circular which is much larger than would occur in practice due to imperfection. The results for hf — OJh are presented to highlight the effects. Practically however, hf = OJDlh represents a more realistic variation in thickness. Convergence studies indicated that for hf*, hf~ = O.lh the use of 30 terms in the solution series (equations 9 and 10) gave 4 significant figure accuracy or better for the frequency factors A* fox k - 0,1, ... 6. This was considered to be acceptable for the purposes of the illustrative examples considered here. In a parallel Finite Element study, beam elements, and two- and three- dimensional plane stress elements used to model an imperfect ring. In order to get 4 significant figures or better, 120 elements were used to model the complete ring. Comparison of the results obtained from the numerical method and the Finite Element Method shows that (i) there is good agreement between the curves of frequency factors obtained by the Finite Element method and the numerical method. (ii) the trends and patterns of frequency splitting are nearly identical irrespective of the analysis methods or the types of finite elements used. In considering the effect of single harmonic variations of the profile of the inner and outer surfaces on the natural fiequencies of different radial modes, the discussion will focus on three aspects: (a) , the effect of the harmonic number of the profile; (b) . the effect of the magnitude of the profile variations; and (c) . the effect of the spatial phasing between the profile variations of the inner and outer surfaces. The frequency splits shown are often very small (~ 0.001%). Note however that in some inertial sensor applications, such small frequency splits may be of practical significance. 1464 (a) The effect of profile harmonic number Table 1 compares the frequency factors A„ obtained for a perfect ring and an imperfect ring for ^-0, "^12, tc, h/" -hj — 0,lh, and i=j = 2 to 6. It is evident from these results that: For the flexural modes {n>2): (i) When i , j are equal and even (see Table 1), frequency splitting only occurs in the nth mode where w = Id. I 2 and /c is an integer. The maximum frequency splitting occurs in the n—H 2 modes (i.e. k ^ 1 ) and the splitting decays as k increases. (ii) When i , j are equal and odd (see Table 1) frequency splitting only occurs in the nth mode where n = ki and k ism integer. The maximum frequency splitting occurs for A: = 1, and splitting decreases as k increases. It should be noted here that frequency splitting in the higher modes exists but is very small, e.g., for <}» = 0, i - j = 2 and kf= OJh , frequency splitting occurs in the 2nd and higher radial modes. It can be seen from Table 1 that the splits in frequency factor are 0.019% at the 2nd mode, 0.001% at the 3rd mode, and less than 0.001% at the 4th mode or higher mode. These correspond to actual frequency splits of about 0.01%, 0.(XK)5% and less than 0.CKX)5% respectively (equation (12)). For the radial extensional mode(« = 0), no frequency splitting occurs. It is clear from Table 1 that the trends and patterns of frequency splitting are the same for ^-0,n/2 and tc. However, frequency splitting is less foT^ = 0 than for <}) = 71 under the same conditions. Frequency splits for ^ between 0 and K are intermediate between those for <{» =0 and <{) = 7t. The above patterns are in agreement with the qualitative results published in reference [5] in which only the conditions for non-splitting are established. 1465 Table 1 The difference of frequency factors A on the radial modes w(n)[the parameters of profile variations are taken as hf=0.1h/ i=j=2 to $ and (a)«|>=0; (b)i^5=TC/2; (c)4>=?c] (a) 4>= =0 A(0) A(2) A{3) A(4) A(5) A(6) Perfect 1.0005 0.003313 0.02645 0.09693 0.2525 0.5406 i= 2 high low -0.011% -0.023% -0.042% -0.041% -0.042% -0,038% -0.036% -0.035% i= 3 high low 0.075% -0.047% -0.088% -0.153% -0.133% -0.108% -0.094% -0.095% i= 4 high low 0.457% 2.155% -2,287% -0.134% -0.150% -0.594% -0.349% -0.240% -0.248% i= 5 high low 1.631% -0.171% -0.141% -0.332% -0.229% -1.816% -0.838% i= 6 high low 5.656% -0.171% 2.406% -2.726% -0.267% -0.715% -0.302% -5’.624% (b) 4)=Tr/2 A(0) A(2) A{3) A(4) A{5) A(6) i= 2 high low 0.007% 0.336% -2.427% -0.595% -0.676% -0.971% -0.972% -1.169% -1.272% i= 3 high low 0.090% -1.231% 0.336% -2.559% -0.721% -0.869% -1.038% -1.040% i= 4 high low 0.382% 14.58% -18.71% -1.150% 0.287% -2.871% -0.868% -0.828% -0.901% i = 5 high low 1.236% -5.642% -1.239% -1.201% 0.212% -3.703% -1.172% i = 6 high low 4.111% -5.620% 13.12% -17.59% -1.316% -1.404% 0.117% -6.370% (c) 41=71 i = 2 high low A{0) 0.026% A(2) 0.690% -4.827% A(3) -1.132% -1.340% A(4) -1,908% -1.912% A(5) -2.308% A{6) -2.514% i = 3 high low 0.106% -2.469% 0.709% -4.934% -1.312% -1.637% -1.989% -1.993% i= 4 high low 0.314% 18.81% -26.96% -2.227% 0.528% -4.993% -1.388% -1.447% -1.532% i= 5 high low 0.881% -10.89% -2.400% -2.135% -0,018% -5.031% -1.510% i= 6 high low 2.808% -10.81% 16.78% -25.36% -2.396% -2.134% -1.881% -5.060% Note: 1. difference = [A (n) - A (n)porfc.ct]x 100% / A (n)perfect i\o 2. A(n) = ——(H (n) , where (ii(n) is the natural frequency at E the nth radial mode. 1466 Table 2. The difference of frequency factors A on the radial inodes w(n) [the parameters of profile variations are taken as hf=0.01h, i=j=2 to 6 and (a)<|>=0; (b)<|)=TC/2; (c)<|)=7t ] (a) (|)=0 Perfect A(0) 1.0005 A(2) 0.003313 A(3) 0.02645 A{4) 0.09693 A(5) 0.2525 A(6) 0.5406 i= 2 high low =0 =0 =0 =0 =0 =0 i= 3 high low 0.001% «0 -0.001% -0.002% -0.001% -0.001% -0.001% i= 4 high low 0.005% 0.222% -0.223% -0.001% -0.001% -0.006% -0.004% -0.002% i= 5 high low 0.016% -0 . 002% -0.002% -0.003% -0.002% -0.019% -0.009% i= 6 high low 0.060% -0.002% 0.256% -0.260% -0.003% -0.008% -0.003% -0-. 063% (b) <j)=7t/2 A{0) A(2) A(3) A(4) A(5) A(6) i= 2 high low =0 0.004% -0.024% -0.006% -0.010% -0.012% -0.013% i= 3 high low 0.001% -0.012% 0.003% -0.026% -0.007% -0.009% -0.010% i= 4 high low 0.004% 1.684% -1.725% -0.011% 0.003% -0.029% -0.009% -0.009% i= 5 high low 0.012% -0.058% -0.012% -0.012% 0.002% -0.038% -0.012% i= 6 high low 0.044% -0.057% 1.550% -1.595% -0.013% -0.014% 0.001% -0.069% (C) <j>=7t i= 2 high low A{0) =0 A(2) 0.008% -0 . 048% A(3) -0.012% A(4) -0.019% A(5) -0.023% A(6) -0.025% i= 3 high low 0.001% -0 . 024% 0.008% -0.050% -0.013% -0.016% -0.020% i= 4 high low 0.003% 2.360% -2.440% -0.021% 0.006% -0.050% -0.014% -0.015% i= 5 high low 0.009% -0.114% -0.023% -0.020% «0 -0.051% -0.015% i= 6 high low 0.028% -0.112% 2.165% -2.251% -0.023% -0.020% -0.018% -0.051% Note: 1. difference = [A (n) - A (n)p*r£ecc]x lOO^s / A(n}p<,r£«ot 2. A(n) = (£t^(n). where tOfn) is the natural frequency of E the nth radial mode. 1467 (b) The elBfect of proBle amplitude The effects of varied profile amplitude (h/' = hf = O.lh and OMlh) upon the frequency factors for ^ = 0, it and i =J = 2 to 6 can be seen by comparing Tables 1 and 2. It may be concluded from these results and others which are presented in [11] that: (1) When n-i 12, frequency factor splitting due to variable profile magnitude compared with the frequency factor of the perfect ring is nearly proportional to the profile amplitudes, hj* and hj . For example, for i ^ = 7 t and hf^ = hf 0,lh ,0.01h (see Tables 1 and 2), the magnitude of frequency splitting at the 2nd mode is 45.77% for hf = OJh and 4.80% for hf= OMlh. These correspond to actual frequency splits of 24% and 2.4% respectively (equation (12)) (2) For modes other than those for which n^i 12, splitting of frequency factors is nearly proportional to the square of the profile amplitudes, and hf . For example, for i “4, 2 nd hf* = hf — O.lh ,0Mlh (see Tables 1 and 2), the magnitude of frequency factor splitting at the 4th mode are 5.521% for hf-O.lh and 0.056% for A/= O.Olh. These results shown in Tables 1 and 2 for (j) = 0 and = ^t/2 show that the general nature of the trends regarding the effect of profile amplitude variations on the frequency factors are the same for all values of ^ , although the magnitudes of the changes in frequency factors depend on ({>, as discussed in the following section. (c) The effect of spatial phase angle variations The effects of the variations of spatial phase angle <j) on the frequency factors are shown in Figure 4, from which it is evident that (1) As frequency splitting occurs (see Figure 4.a-4.d ), the maximum frequency splitting is obtained at <) = tc and the minimum splitting occurs at (p = 0. It is clear that the maximum frequency splitting occurs in the n — il2 modes. (2) In modes for which no frequency splitting occurs (see Figure 4.e and 4.f), the minimum frequency difference compared with that of the perfect ring is detected at <j) = 0 and the maximum at ^ = n. Irrespective of the value of (j) , the frequencies of these modes are always less than the corresponding frequencies of the perfect ring. 1468 WKemnceIn 4 (%) at(eiwic»ln A (%) Figure 4. Effect of spatial phasing on frequency factors with hf = O.lh and (a) i=j=:4,2nd mode; (b) i=:j=6,3rd mode (c) i=j=2, 2nd mode ; (d) i=j=3,3rd mode (e) i=j=3, 4th mode ; (f) i=j=4, 5th mode 1469 4, Conclusions In this paper, Novozhilov thin-shell theory and the Rayleigh-Ritz procedure have been applied to derive the frequency equations of a thin ring with a rectangular cross-section and a circumferential proffle variatioa Profile variations are represented, in the general way, by Fourier series functions and the method gives quantitative predictions of frequency splitting. The observed firequency splitting patterns are in agreement with previously published qualitative results. Numerical results have been presented for example cases in which the inner and outer profiles are nominally circular with superimposed single-harmonic variations m radius. The effects on frequency splitting of the harmonic number of the profile variation, and the amplitude and spatial phasing between the inner and outer surfaces have been investigated. References 1. Love, A-E.H., A Treatise on the Mathematical Theory of Elasticity^ Dover Publications, New York, fourth edition, 1952. 2. Fox, A simple theory for the analysis and correction of frequency splitting in slightly imperfection rings. Journal of Sound and Vibration, 1990,142(2), 227-43. 3- Fox, Vibrating cylinder rate gyro: theory of operation and error analysis. Proceeding of DGON Symposium on Gyro Technology, Stuttgart, 1988, Chapter 5. 4. Chamley, T. and Perrin, R., Studies with an eccentric bell. Journal of Sound and Vibration, 1978, 58(4), 517-25. 5. Perrin, R., Selection rules for the splitting of the degenerate pairs of natural frequencies of thin circular rings. Acustica, 1971, 25, 69-72. 6. Chamley, T. and Perrin, R., Perturbation studies with a thin circular ring. Acustica, 1973, 28, 139-46. 7. Valkering, T.P. and Chamley, T., Radial vibrations of eccentric rings. Journal of Sound and Vibration, 1983,86(3), 369-93. 8. Tonin, R.F. and Bies, D.A., Free vibration of circular cylinders of variation thickness. Journal of Sound and Vibration, 1979, 62(2), 165-80. 9. Novozhilov, V.V., The Theory of Thin Shells, P. Noordhoff Ltd., The Netherlands, 1959. 10. Kreyszig, E., Advanced Engineering Mathematics, John Wiley & Sons, Inc., Singapore, 1993, pp. 569-71. 11. Hwang, R., Free vibrations of a thin ring having circumferential profile variations. Ph.D. Thesis, University of Nottingham, U,K, (in preparation) 1470 FREE VIBRATION ANALYSIS OF TRANSVERSE-SHEAR DEFORMABLE RECTANGULAR PLATES RESTING ON UNIFORM LATERAL ELASTIC EDGE SUPPORT D.J. Gorman University of Ottawa 770 King Edward Ave., Ottawa, Canada KIN 6N5 ABSTRACT Utilizing the Superposition Method a free vibration analysis is conducted for transverse-shear deformable rectangular plates resting on uniformly distributed lateral elastic edge support. Edges are free of moment. The thick isotropic Mindlin plate is utilized for illustrative purposes. The Mindlin equations are satisfied throughout. Typical computed results are plotted for a square plate. INTRODUCTION It is well accepted that classical rectangular plate boundary conditions denoted as simply supported or clamped are often not achieved in real structures. This is because of elasticity in the edge supports. Furthermore, in many rectangular plate installations elastic edge supports may be utilized intentionally. For this reason a number of studies of effects of elasticity in the edge supports on rectangular plate free vibration frequencies have been conducted and results published. Almost all of these studies have been devoted to the free vibration behaviour of thin isotropic plates. Studies by the author, related to this family of vibration problems, have been devoted to situations where elastic stiffness is uniformly distributed along the edges as well as cases where the stiffnesses are arbitrarily distributed. All of his studies have been conducted by means or the Superposition Method and in a fairly recent article he has demonstrated that all of these families are amenable to analytical type solutions [1]. In this paper we exploit the powerful Superposition Method to analyse the free vibration behaviour of transverse-shear deformable plates resting on uniform lateral elastic edge support. This represents a much more complicated problem than the thin isotropic plate problems discussed above. For our purposes we choose the thick shear-deformable Mindlin plate and base our solution on Mindlin theory. In the interest of keeping the literature review up to date the recent publication 1471 of SAHA, KAR, and DATTA [2] is drawn to the attention of the reader. They report on a study of thick Mindlin plates resting on edge supports with uniform lateral and rotational elasticity. They have employed a Rayleigh-Rite energy approach. Plate lateral displacement is represented by a rather complicated set of Timoshenko beam functions, each extremity of each beam being attached to a local lateral and torsional spring. It will be seen that no such functions need be selected in the superposition approach adopted here. Another related paper is one by the present author dealing with Mindlin plates where lateral displacement along the plate edges is forbidden but uniform rotational elastic support is provided [3]. This problem is somewhat easier to solve since edge lateral displacement is forbidden and, unlike the present problem, mixed derivatives do not show up in the boundary condition formulation. This latter problem was shown to be amenable to solution by the modified Superposition-Galerkin Method which is extremely easy to use when it is applicable. MATHEMATICAL PROCEEDURE A solution to the present problem is obtained through th& supei-position of the eic^ht edge-driven forced vibration solutions (building blocks) shown scheraaticaly in Figure 1. All of the non-driven edges have slip-shear support. This type of support, indicated in the figure by two small circles adjacent to the edc^e, implies that the edge is free of torsional moment and transverse shear fomes. Furthermore, rotation of the plate cross-section along the edge is everywhere zero. We begin by examining the first building block. Its driven edge is free of torsional moment, and rotation of the plate cross-section along this boundary is every where zero. This latter condition is indicated by two solid dots adjacent to the edge. Driving of this edge is accomplished by a distributed harmonic transverse sheai* force of circular frequency o). The spatial distribution of the shear force is expressed as, Q,l,.,= E E„cos(m-l)7t5 ( 1 ) m = 1,2 where k is the number of terms required in the series. 1472 Fig. 1 Schematic representation of building blocks utilised in theoretical analysis. We now examine the response of the above building block to this harmonic excitation. The proceedure followed is almost identical to that described in an earlier publication [4]. A concise description will be provided here for the sake of completeness. The governing differential equations which control the response of thick Mindlin plates are written in dimensionless form as, W d" W ^ ^ ^ W = 0 (2) 8 ^“ (j)- art“ ({) ari V 3 d^- 4)- 8 Ti^ 4) 5 ^ 8 r| i ^ 5 U (3) 8f 4)^v, a-n- 4)Vi d^dr\ 1473 Transverse shear forces, bending moments, etc., are written as. V 9 „ ^ 3^ cj) 3r| ^ 3 ti (j) 3n at a tf 9 <|i- 1 a i(r. When subjected to the first term of the driving force (Eqn 1) the response of the building block will be essentially that of a Timoshenko beam. The governing differential equations reduce to a set of two which may be written as d“W ^ d-ijj A'^(})“c|);;W -+ (p-!• +- d rt- d Ti"“ V3 = 0 (6) and d'l};^ V3<{)V ^ 1 dW K ^2 dn j 12 (7) It is convenient to represent the lateral displacement W, and plate cross- section rotation as, W(Ti) = X(Ti), and (ti) = Z (t)) The governing differential equations may then be written as X"(q) + a^,Z'(q) + b„,X(ti) = 0 (8) and Z''(q) + a,^3X'(n) + b„,5Z(Ti) = 0 (9) where superscripts imply differentiation with respect to r|. Coefficients 1474 .... etc,, are defined in reference [4]. Applying the appropriate differential operators to this set of equations the parameter X (ti) is eliminated and a second order ordinary homogenous differential equation is obtained involving the parameter Z (q). It is found that for our range of interest the roots of the characteristic equation associated with this differential equation are always real. There are then three possible pairs of roots depending on the coefficients in the above differential equation. Designating these pairs of real roots as Rj and possible forms of solution exist as follows, Casel, RpR2<0-0 Case2, R,<0-0; R,>0-0 CaseS, Ri,R2>0-0 (lO) In all work reported here it has been found that only, case 2, has been encountered. It will be obvious that the functions X (q) and Z (q) must be symmetric with respect to the ^ axis. We may therefore write for case 2, X(q) = cos aq + cosh pq, (11) and Z(^) = Am ( cos a q + cosh P q (12) where a = ^|R,| , and P = Expressions and S^.^re obtained by taking advantage of the coupling of equations 8 and 9, as was done in Reference [4]. We then impose the boundary conditions, Q^ = E,^^,and ilf^ = 0,at q=l, in order to evaluate the unknowns and B^^^ of Equation (11). Accordingly we obtain, E X(q) = .^{cos aq + X 1 cosh Pq}, (13) and 1475 Z(T 1 ) = ^ {S„, sin a n + XI S„, sinh P t|}, (14) where XI and X2 are easily evaluated. Next we examine the response of the first building block to driving terms where m>l. We follow the proceedure described in Reference [4]. Levy type solutions for the parameters W, .... . etc., are written as. W(5,ti) = XjTi) cosirni? (15) litres,Tl) = Y„(ri) sinm-ii^ • (16) = Z„(T|) cosmn? (17) It will be noted that all required boundary conditions along the edges, ^ = 0, and ^ = 1/ are satisfied. Next, the above expressions (Eqns 15,16,17) are substituted in the set of governing differential equations. The following set of coupled ordinary differential equations, written in matrix form are obtained X 0 0 X./' t>n,l ^m2 0 x„ o' Y ^ m 0 0 ^m2 Y.' ^m4 0 Y„, ►_ 4 0 zj . m J .^m3 ^m4 0 0 0 bn,5. 0 (18) Again, the quantities .... etc., are defined in Reference [4]. Applying the appropriate operators on the above equations, as was done in Reference [4] we are able isolate a single homogenous sixth order ordinary differential equation involving the dependent variable (ti), only. Because first, third, and fifth order derivates are missing from this equation the associated characteristic equation can be formulated as a cubic algebraic equation. Again it is found that for the range of the 1476 present study all of the roots are real Designating these roots^ as R ,R,, and R3, it follows that four solution cases are possible depending orl the coefficients of the characteristic equation. They are. Case!, R,, R,, and R. < 0-0 Case2, R,, R^ < 0-0 ; R3 > 0-0 ‘ - (19) Cases, Rj < 0-0 ; R, and R 3 > 0-0 Case4, Rp R, and R 3 > 0-0 Inthepresentstudyonlycase3,andcase4are€ncountered.Introdudng a =^\Rj 1 , P = , and y = JIR 31 and recognizing that (q)must be antisyrnmetric about the ^ axis while X^(q)and Y^(q)must be symmetric, we are able to write for case 4, Y^(q) = A,^ cosh aq+B^^ cosh Pq + C^^^ cosh yq ( 20 ) Utilizing the coupling of the ordinary differential equations, as in Reference [4], it follows that we may write, X^/q) = A^R,^i cosh ocq + B^R^, N + (21) and Z^(q) = A^S^, cosh ocq + B^^S^, cosh Pq+C,^iS ^3 cosh yq (22) The quantities R^,,, S^p ... etc., are evaluated following steps described in Reference [4]. Expressions for (q), X^^ (q), etc., for case 3 will differ from the above expressions only iii that Cosh a q must be replaced by cos a q. The unknown constants A^, B „p etc., of the above solutions are evaluated by enforcement of boundary conditions along the edge, q=l. These conditions comprise zero torsional moment, zero edge rotation, with transverse shear force = E^p For case 4 we obtain (n) = — {cosh a q + X1 cosh p q + X 2 cosh y q} (23) with the functions X^(q) and (q) differing from Y,^(q) only in that 1477 R^j and S^j, etc., must be included. We therefore now have the exact response of the first building block to the imposed driving force components available. It will be observed in Figure 1 that the second, fifth, and sixth building blocks differ from the first only in that they are driven along different edges. Solutions for their response are therefore easily extracted from that of the first. Focusing our attention on the third building block we find that its driven edge is free of transverse shear forces and torsional moment. It is driven by a distributed cross-section harmonic rotation. The spacial distribution of this imposed driving rotation is also represented by the series of Equation 1. The reader will appreciate that a solution for the third building block is obtained by following steps identical to those described for the first. Only the imposed boimdary conditions along the driven edge differ. Solutions for the quantities W, etc., will be identical in form to those already developed for the first building block except that quantities XI, XI, etc., will be slightly different. We designate them as XIP, X2P, etc., for the edge-rotation driven building blocks. Solutions for the remaining four building blocks of Figure 1 are therefore available. THE EIGENVALUE MATRIX This matrix is shown schematicaly in Figure 2. It is generated following established practices. Let us first consider the transverse force equilibrium condition along the edge, rj = 1. It is readily shown that this equilibrium condition is written in dimensionless form as, Q, + K„W = 0 (24) The plus sign of this equation must be replaced by a minus sign when we formulate the corresponding equations for the edges, 11 = 0 , and ^=0. 1478 Em 1 2 3 ^ En 1 2 3 c:> CM C'> - W 4 ^ Ep i 2 3 - 1 10 ^ E, 1 2 3 r&ean-, % < ^ Es 1 2 3 1 2 3 - - - - ■ - ::: - : : : LvJ : : : - - - - - _ - - - : : : - - « - - - Ill - : : : - 1 1 M 1 t p- - ^ - - - - - : : : : : : - □ - : : : - - - : : : *_ [v + r _ - - - - - - - - “ " - : : : ■ - : : : - : : : Ill - : : : 1 M 1 i" j ]' : : : - : : : - " " I ■ I I - Pig 2 Schematic representation of Eigenvalue matrix based on three-term function expansions. Short bars indicate non¬ zero elements. M or V on inserts to right indicate edges along which moment or lateral force equilibrium is enforced. To construct the first three equations upon which this matrix is based we superimpose all eight building blocks and expand their net contribution to displacement W in a cosine series. The transverse shear force along the edge, 11=1, is already available in such a series. We then express the left hand side of equation 24 in series form and require that each net coefficient in this series must vanish. This leads to 3 homogenous algebraic equations relating the 8 k imknowns where, for the illustrative matrix of Figure 2, k equals 3. A second set of three homogenous algebraic equations is obtained by enforcing the corresponding lateral equilibrium condition along the edge. 1479 ^ = 1. Moving down the matrix of Figure 2 it is seen that a third and fourth set of equations are obtained by enforcing the moment equilibrium condition, i.e., net bending moment equals zero along the same edges, in an identical fashion. Finally, it is seen in Figure 2 that four more sets of equations are obtained by enforcing the required equilibrium conditions along the edges, ti=0, and ^=0. We thus have, in general, 8 k homogenous algebraic equations relating the 8 k unknown driving coefficients. The coefficient matrix of this total set of equations forms our Eigenvalue matrix. Certain measures can be taken to greatly simplify and expedite generation of the matrix. It will be observed (Fig. 2) that the matrix is composed of 64 natural segments. This array of segments may be referred to through the indices (I, J). It is expedient to first generate the matrix without including contributions related to the driving shear forces along the building block edges (Eqn. 24). The matrix is then completed by adding the quantity 1.0 to diagonal elements of segments (1,1) and (2,2), and subtracting 1.0 from the diagonal elements of segments (5,5) and (6,6). Physical reasoning leads also to another vast signification. One may begin by generating the elements of the matrix lying below the first four building blocks, only, (Fig. 2). Following a proceedure as discussed in Reference [4], and exercising caution with respect to necessary sign changes, all of the remaining segments of the matrix may be extracted from those already generated. Eigenvalues are, of course, those values of the dimensionless frequency, X-, which cause the determinant of the Eigenvalue matrix to vanish. Mode shapes are obtained after setting one of the non-zero driving coefficients equal to zero and solving for the others. PRESENTATION OF COMPUTED RESULTS It will be appreciated that problems involving vast arrays of stiffness coefficients, plate aspect ratios, thickness-to-length ratios, etc. can be resolved by the proceedure described above. Only a single typical problem and its solution will be discussed here, for illustrative purposes. We consider a square plate with equal dimensionless lateral elastic 1480 stiffness imposed along each edge. Results are presented for two thickness-to-length ratios, 0.01, and 0.1. Two important observations may be made before examining these results. First, for the very thin plate of thickness-to-length ratio, 0.01, we expect the Eigenvalue vs edge- stiffness ratio curves to almost co-inside with those for a thin isotropic plate based on thin plate theory. Secondly, we recall that the Eigenvalue limits for a thin plate will equal 0.0, and 2 as the elastic stiffnesse approaches natural limits of 0.0, and infinity. Results of a free vibration study of the above plate are presented in Figure 3. It will be noted that computed Eigenvalues are plotted against the parameter Kli / (j)J. By presenting data in this manner it is found 1481 that Eigenvalues for both the thick and the thin plate can be plotted on the same Figure. The absissa of the figure appears in five logarithmic decades. This range has been selected with a view to providing information for the